Section 1
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subtracting polynomials
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Section 1
(50 cards)
subtracting polynomials
change the signs of the terms being subtracted, then add ex: a-b = a+(-b)
zero exponent
a(0) = 1, as long as a is not 0
power of quotient rule
if n is a positive integer and a & c are real numbers, then (a/c)(n) = a(n)/c(n), and c does not equal 0
A linear equation is...
variables x_1, x_2,...,x_n, any equation that can be written in the form: a_1x_1+a_2x_2+...+a_nx_n=b; a_1, a_2, ..., a_n are real numbers
GCF of list of common variables raised to powers
the variable raised to the smallest exponent in the list.
FOIL method
F: product of FIRST terms O: product of OUTER terms I: product of INNER terms L: product of LAST terms; then combine like terms
degree of a term
the sum of exponents on the variables contained in the term
substitution method
substituting 2nd equation into 1st equation
perfect square trinomial
a trinomial that is the square of some binomial: x(sq)+4x+4= (x+2)sq formulae: 1) a(sq)+2ab+b(sq)=(a+b)(sq); a(sq)-2(ab)+b(sq)=(a-b)(sq)
Gauss elimination method
1) keep the x_1 term in one equation, eliminate x_1 from all others 2) keep the x_2 term in the second equation, eliminate x_2 from all next ones 3) and so on
quotient rule for exponents
if m & n are positive integers & a is a real number, then a(m)/a(n)=a(m-n) as long as a is not 0
Solving a linear system:
Replace the system by an equivalent system that is easier to solve
squaring a binomial
= to the square of the first term plus or minus twice the product of both terms plus the square of the second term (a+b)(sq) = (a)(sq) + 2ab +2b(sq) OR (a-b)(sq) = a(sq)-2ab+b(sq)
product rule for exponents
if m and n are positive numbers, and a is a real number, then a(m) * a(n) = a(m+n) {add exponents but keep common base}
Two linear systems are equivalent if...
they have the same solution set
Solving a two variable system amounts to...
finding the intersection of two lines
to solve quad expressions by factoring
1) write the equation in standard form: ax(sq)+bx+c=0 2) factor the quadratic 3)set each factor containing a variable = to 0 4) solve the equations 5)check in the original equation
Any linear system can have either...
1) no solutions 2) exactly 1 solution 3) infinite many solutions
power rule for exponents
if m and n are positive integers and a is a real number, then multiply exponents and keep the base a(m)(n)= a(mn)
standard form of quadratic equations
formula: ax(sq)+bx+c=0, with a not being 0 ex: x(sq)=16 in standard form is x(sq)-16=0 y=2y(sq)+5 in standard form is 2y(sq)+y-5=0
zero factor theorem
if a & b are real numbers and if ab=0, then a=0 or b=0 ex: if (x+3)(x-1)=0, then x+3=0 or x-1=0
slope of a line
the slope(m) of a line containing points (x(1), y(1)) & (y(1), y(2)) is given by: m = rise/run = y(2)-y(1)/x(2)-x(1) as long as x(2) does not equal x(1)
sum or difference of cubes
a(cu)+b(cu) = (a+b)(a(sq) -ab+b(sq)) ex: y(cu)+8= y(cu)+2(cu) =(y+2) (y(sq)-2y+4) OR a(cu)-b(cu)= (a-b)(a(sq)+ab+b(sq) ex: 125z(cu)-1= (5z(cu)-1(cu)= (5z-1)(25z(sq)+5z+1)
finding slope given 2 points on a line
slope= change in y (vertical change)/change in x (horizontal change) to do this, choose two points of a line. label the two x-coordinates of two points x(1), x(2) [read "x sub one" and "x sub 2], and label the corresponding y-coordinates y(1) and y(2) the vertical change(rise) btwn these points is the diff in the y-coordinates: y(2)-y(1). the horizontal change(run) btwn the points is the diff of the x-coordinates: x(2)-x(1). the slope of the line is the ratio of y(2)-y(1) to x(2)-x(1), and we use the letter m to denote slope in this formula: m=y(2)-y(1)/x(2)-x(1)
to graph x<or=3
shade the numbers to the left of 3 and place a bracket at 3 on the number line. the bracket indicates that 3 is a solution:3 is less than or =to 3. in interval notation we write (-infinity,3] *may be easier to graph the inequality first then write it in interval notation. to help, think of the number line as approaching -infinity to the left or +infinity to the right. then simply write the interval notation by following your shading from left to right
problem-solving linear equations
1) understand the problem. read and re-read it. choose 2 variables to represent the two unknowns. construct a drawing if possible. propose a solution and check. 2) translate the problem into two equations 3)solve the system of equations 4) interpret the results: CHECK the proposed solution in the stated problem and state your conclusion
Scaling:
multiply all terms in one equation by a nonzero constant
factoring trinomials
1) use the form x(sq) + bx + c 2) factor out the GCF and then factor a trinomial of the form x(sq) + bx + c To factor ax(sq) + bx + c, try various combinations of factors of ax(sq) and c until a middle term of bx is obtained when checking
power of a product rule
if n is a positive integer and a & b are real numbers, then (ab)(n) = a(n)b(n)
A solution to the linear system x_1, ..., x_n is...
any ordered n-tuple (S_1,...,S_n) of real numbers if you substitute S_1 for x_1, S_2 for x_2, ..., S_n for x_n, then every equation becomes a true relation
A matrix is...
a rectangular array of numbers aligned in rows and columns
multiplication property of inequality
1) if a,b,c are real numbers, and c is positive, then a<b & ac<bc are equivalent inequalities 2) if a,b,c are real numbers, and c is negative, then a<b & ac>bc are equivalent numbers. *the direction of the inequality symbol must be reversed for the inequalities to remain equivalent
degree of a polynomial
the greatest degree of any term of the polynomial
difference of squares
a(sq)-b(sq) = (a+b) (a-b) ex: x(sq)-9=x(sq)-3(sq) = (x+3)(x-3)
slope-intercept form
y=mx+b
Replacement:
add a multiple of one equation to another
interval notation
instead of open circle, parenthesis are used. instead of closed circle, brackets are used. -infinity is all the numbers less than x to infinity.
multiplying the sum & diff of two terms
the product of the sum & difference of two terms is the square of the first term minus the square of the second term (a+b) (a-b) = a(sq)-b(sq)
factor trinomials by grouping
ax(sq)+bx+c 1) find two numbers whose product is a*c and whose sum is b 2) rewrite bx, using the factors in step 1 3) factor by grouping ex: 3x(sq)+14x-5 step 1: 15&-1 step 2: 3x(sq)+14x-5 = 3x(sq)+15x-1x-5 step 3: 3x(x+5) -1(x+5) = (x+5) (3x-1)
solving linear inequalities in one variable
1) clear the inequality of fractions by mult. both sides of ineq. by LCD of all fractions in the inequality. 2) remove grouping symbols such as ( ) by using distributive property 3) simplify each side of inequality by combining like terms 4) write the inequality w variable terms on one side and numbers on the other by using addition prop of inequalities 5) get the variable alone by using multiplication prop of inequalities
factor by grouping
1) arrange terms so the first two terms have a common factor & the last two have a common factor 2) for each pair of terms, factor out the the pair's GCF 3) if there is now a common binomial factor, factor it out 4) if no common binomial factor, begin again, rearranging terms differently. if no rearrangement works it can't be factored.
graphing
picturing the solutions of inequalities on a number line. the picture is called the graph
Solving a linear system means...
to find the solution set, i.e. the set of all possible solutions
A system of linear equations (a linear system) is...
x_1,x_2,...,x_n a collection of linear equations that is the value of the same set of variables
linear inequality in 1variable
an inequality that can be written in the form ax+b<c where a,b,c are real numbers and a is not 0
adding polynomials
combine all like terms
addition property of inequality
if a, b, and c are real numbers, then a<b & a+c<b+c are equivalent inequalities
Interchange:
swap 2 equations
solving linear equalities in one variable
1) clear the equality of fractions by mult. both sides of ineq. by LCD of all fractions in the equality. 2) remove grouping symbols such as ( ) by using distributive property 3) simplify each side of equality by combining like terms 4) write the equality w variable terms on one side and numbers on the other by using addition prop of equalities 5) get the variable alone by using multiplication prop of equalities
linear equation in 3 variables
1) write ea. equation in standard form: Ax+By+Cz=D 2)choose a pr of equations& use the equations to eliminate a variable 3) choose any other pr of equations & eliminate the same variable as in step 2 4) two equations & two variables should be obtained by step 2 & step 3. solve this system for both variables 5) to solve for the 3rd variable, substitute the values of the variables found in step 4 into any of the original equations containing the third variable 6) check the ordered triple solution in all three orig. equations
Section 2
(50 cards)
Two matrices A and B are row equivalent if...
there is a sequence of row operations that transform one into another
A matrix with one column is called a...
column vector
Verticle line test:
Given the graph of a relation, if you can draw a verticle line that crosses the graph in more than one place, the relation is not a function
T is onto if and only if...
the equation Ax=b is always consistent, the matrix A has a pivot in every row, the column of A spans R^m
Symmetric about the Y-axis:
Whatever the graph is doing on one side of the Y-axis is mirrored on the other side
The set of all possible linear combinations of v_1, ..., v_p is called...
the subset of R^n spanned by v_1, ..., v_p, denoted by Span(v_1,...,v_p)
The vector equation x_1a_1+x_2a_2+...+x_na_n = b and the system with the augmented matrix [a_1,a_2,...a_n|b] are equivalent if and only if
the corresponding system is consistent
The identity matrix is...
I_n, nxn matrix
To compute the i-entry of Ax...
multiply every entry in the i-row of A by corresponding entries in x, take the sum of these products
The arguement of a function:
Is the " X " in f(x)
The leading entry of a nonzero is...
the leftmost nonzero entry
If v_1,v_2,..,v_pER^n then the set {v_1,v_2,v_p} is dependent if and only if
one of the vectors is a linear combination of the preceding ones
R is...
the set of all real numbers
If the system is consistent, these statements are true:
1) The equation Ax=b is always consistent for all bER^m 2) The column a_1, a_2,...,a_n fo A span R^m 3) the coefficient matrix has a pivot in every row
Relation:
A relationship between sets of infomation
The free variable is...
the non-pivot column of the coefficient matrix
v_1...v_p are linear dependent if...
there is a nontrivial linear relation between them; i.e. fi we can find some scalars c_1, c_2, ..., c_p, not all zeros, where c_1v_1 + c_2v_2 +...+c_pv_p = 0
T is 1-1 if and only if...
the homogeneous equation Ax=0 has only the trivial solution, has no free variables, has a pivot in every column, the n column of A are linear independent
onto
range (-infinity, infinity)
Augmented matrix transformation is...
linear
The product Ax = x_1a_1+x_2a_2+...+x_na_n and is only defined when...
the number of columns in A equals the number of entries in x
Fundamental question about a linear system:
Is the system consistent, i.e. does a solution exist, if it does, is this a unique solution?
The basic variables are...
the variables corresponding to the pivot column in the coefficient matrix
Any given matrix A is row equivalent to...
exactly one matrix B in Reduced Echelon form; B = rref(A)
R^n is...
the set of all vectors in n entries with real coefficients
v_1,v_2,...,v_p are independent if and only if...
the system has no free variables
Best way to solve any system of linear equations is to...
row reduce the augmented matrix
The system Ax=b is called homogeneous if...
b=0
f(x) = X² + 2 :
Shifts parabola up two units
A linear system is consistent if and only if
the last column of the augmented matrix is not a pivot column, a consistent linear system has 1 solution precisely when it has no free variables
A transformation T:R^n-R^m is linear if...
T respects the vector addition and scalar multiplication; i.e. T(u + v) = T(u) + T(v), T(cu)=cT(u)
Elementary row operation:
row replacement, row interchange, row scaling
How to solve a linear system
1) Write down the augmented matrix B of the system 2) Row reduce B to Echelon form; if the last column of B is pivot, the system is inconsistent; if the last column of B is not pivot, it is consistent 3) Complete the row reduction to C = rref(B) 4) Write down the new system corresponding to C; locate basic/free variables 5) Use the new system to express all variables in terms of free variables
The homogeneous system Ax=0 has a nontrivial solution if and only if...
the system has infinite many solutions, the system has at least one free variable
If you have more vectors than the number of entries in each of them, i.e. p>n then they are...
dependent
Any row/column is...
a row/column with 1 nonzero entry in it
To find the parametric vector form for the solution set...
1) Row reduce the augmented matrix 2)Get a parametric description of the solution set 3) Write your solution x as a column vector; single out the coefficient for each variable in a column
v_1...v_p are linear independent if...
there is no non-trivial linear relation between them, i.e. has only the trivial solution
The system Ax-b is not homogeneous if...
b!=0
1-1
same images
Unordered lists:
Are sets of information; {2, 4, 8}
The homogeneous system is always...
consistent, x=0 is always a solution
Row reduction algorithm
1) Locate the first pivot column/first pivot position 2) Locate a pivot, choose a nonzero entry in the pivot column to be the pivot, if necessary do row interchange to move the pivot to the pivot position, if necessary do row scaling to make the pivot equal to 1 3) Create zeros below the pivot use row replacement to make all entries below pivot equal to 0 4) Cover the row containing the pivot and any row above it, apply steps 1-3 to the remaining submatrix 5) Start from the right most pivot and then move upward and to the left, if a pivot does not equal 1, make it equal 1 with row scaling, use row replacement to create zeros above each pivot
A is in Echelon from if it has 2 properties:
1) All nonzero rows have to lie above any row of all zeros 2) The leading entry of any nonzero row is to the right of the leading entry of any row above it
x=0 is the...
trivial, zero solution
The operation of reading any vector vER^n to p + v is called...
the translation by p
A is in Reduced Echelon form if...
1) All nonzero rows have to lie above any row of all zeros 2) The leading entry of any nonzero row is to the right of the leading entry of any row above it 3) The leading entry of any nonzero row is 1 4) The entry is the only nonzero entry in the column
f (x) :
Means plug a value for " X " in a formula " f "
v_1, v_2, ..., v_p are dependent if and only if...
the system has >= 1 free variable
f(x) = X²:
Parabola pointing up; graphs at zero
Section 3
(50 cards)
ⁿ√60 where n=6:
60 1/6
f(x) = | X - 3 | :
Shifts V three units right
f(x) = | X | - 3 :
Shifts V down three units
f(X) = | X | + 1 :
Shifts V up one unit
f(x) = - √ x :
Graphs +x, -y to infinity
Slope-intercept form:
Y=mx + b
f(x) = 4 X² :
Shrink of a parabola
Reflection about the "X" axis:
The - sign outside the arguement indicates what?
f(-x) :
Example of a reflection about the y axis
(f/g)(x)= :
f(x)/g(x)
Linear Function formula:
f(x)=mx + b ; where m ╫ 0
f(x) = ( X-3 )² :
Shifts parabola right 3 units
Are the inverse of each other:
√ and ( )²
f(x) = ¼ X :
Stretch of a parabola
f(x) = √ -x :
Graphs -x, +y to infinity
- f(x):
Example of a reflection about the X axis
f(x) = X² - 3 :
Shifts parabola down three units
Union symbol:
U
f(x) = | X | :
V pointing up
f(x) = ( X + 3 ) ³ :
Shifts graph left 3 units
f(x) = X³ - 5 :
Shifts graph down 5 units
f(x) = ±√ X :
Not a function because it would have two answers; + 9 or - 9
Horizontal line test:
A function is one-to-one ONLYif every horizontal line intersects the graph at ONLY one point
f(½x):
Example of a horizontal stretch
√5 √5 = :
√ 5 × 5
f(X) = - | X + 7 | :
Reflection (V pointing down) shifted seven left
2f(x):
Example of a vertical stretch
³√ a = :
a 1/3
Intersection symbol:
∩
One to One function:
No two ordered pairs in a function have the same second componet (y-value)
g ° f = :
g(f(x))
f(x) = - X³ :
Reflection of X³
f(2x):
Example of a horizontal shrink
Point slope form:
y - y1 = M ( X - X1 )
Slope of a Line:
M= Y2 - Y1 / X2 - X1 , where X1 ╪ X2
f(x) = | X + 2 | :
Shifts V two units left
f(x) = √ X :
Graphs as ½ of a horizontal parabola pointing towards ∞
f ° g = :
f (g (x))
Reflection about the "Y" axis:
The sign inside the arguement indicates what?
½ f(x) :
Example of a vertical shrink
11x/2-x divided by 11/2-x:
11x/2-x × 2-x/11 = ?
f(x) = ( X - 6 ) ³ :
Shifts graph right 6 units
f(x) = X³ + 3 :
Shifts graph up 3 units
f(x) = - √ -x :
Graphs -x, -y to infinity
f(x) = X³ :
A vertical line that bends to the right, goes horizontal, then bends to the left, turning vertical again; this crosses at the (0,0)
f(x) = ( X + 2 ) ² :
Shifts parabola left 2 units
Constant function:
f(x)= mx + b , where m = 0
a/b divided by c/d:
a/b × d/c = ?
Linear graph in 2 variables:
A + By = C , where A or B ╪ 0
Exponet rule:
√16+9 ╪ 4 + 3
Section 4
(51 cards)
How would you describe this inequality in a solution set: x < 5 ?
{x|x<5}
What is the difference in slopes for perpendicular lines?
They have negative reciprocals. ex. Line¹: m= 2/4, would mean Line²: m= -4/2
What do parallel lines have in common?
Equal slopes. m¹ = m²
The graph of y = b is?
A horizontal line through b on the y axis. (Slope = 0)
How does slope tell us the tilt? y/x
Because y/x = rise/run
These two equations would be easiest to solve by what method Equation¹: y = -2x + 6 Equation²: 3x - 2y = 16?
The substitution method, substitue first equation in place of y in the second: 3x - 2(-2x + 6) = 16 and solve.
If m=0 what type of line would be viewed on the graph?
A horizontal line.
How is a positive line viewed?
It runs upwards. m>0
What is the first step in this equation : 2|2x+1| -4 = 16 ?
Get rid of the variables outside of the absolute value.
What three methods could be used to solve systems of linear equations? Equation¹: 2x + y = 6 Equation²: 3x - 2y = 16
By graphing, the substitution method, and the addition method.
For what inequality symbols do you use parenthesis?
< and >
Horizontal line:
Y = b ; slope 0
How is a negative line viewed?
It runs downwards. m<0
How would you express this inequality in a solution set: -4 < t ≤ 5/3 ?
{t| -4 < t ≤ 5/3}
What will the solution of this absolute value equation look like |exp.| < a ?
-a < exp < a
What is the solution for |x - 5| < 0?
∅ The absolute value of a number can never be < 0. No solution.
How do we view lines on the graph to get the correct slope?
Left to right
What do you do when you have a negative denominator? ex b-2/-5
Move it up top. answer: 2-b/5 (Make sure to change signs)
Is this a linear equation or a linear function: f(x) = 2x - 1 ?
A linear function
How do you find the equation of a line if you are given two points, but no y-intercept or slope? Such as: (4 , 3) (6 , -2)
You would have to find the slope first, x² - x¹/y² - y¹, and then use the point slope equation, y - y¹ = m(x - x¹)
For what inequality symbols do you use brackets?
≤ and ≥
When graphing linear inequalities, for what symbols would you use a dotted line in the graph?
< and >
What is the first step in graphing a linear inequality? y ≥ 1/2x - 3
Decide what type of line it will be. In this case it's a solid line because of the ≥ sign.
What is the equation of the line through (0 , 2) that is perpendicular to the graph of the line: y = 1/2x - 4 ? (Think of the perpendicular rule)
y = -2/1x + 2 because (0 , 2) gives us the y intercept and -2/1 is the reciprocal of the line in the question.
What will the solution of this absolute value equation look like |exp.| = a ?
exp = a or exp = -a
Is this a linear equation or a linear function: y = 2x - 1 ?
A linear equation.
What is the formula for slope-intercept form?
y = mx + b
How would you word the function (front) part of this equation? P(x) = 7.25x
P is a function of x
What is the point slope equation? Used when you have a slope and a random point.
y - y¹ = m(x - x¹)
How is slope defined?
The slope of a line measures the tilt of the line.
These two equations would be easiest to solve by what method Equation¹: 4x + 2y = 6 Equation²: 3x - 2y = 16?
The addition method. If set up as if adding them together the y's would cancel. Then you could solve for x and then solve for y.
What will the solution of this absolute value equation look like |exp.| > a ?
exp. < -a or exp. > a
Vertical line:
X = s ; slope undefined
How do you graph the line of a linear inequality? (Also the 2nd step) y ≥ 1/2x - 3
Set the inequality as an equation y ≥ 1/2x - 3 turns into y = 1/2x - 3, and plot the line.
When graphing linear inequalities, for what symbols would you use a solid line in the graph?
≤ and ≥
A linear system where two lines cross (one answer) would be what type of system?
A consistent system.
What is the standard form for a linear equation?
ax + by = c Both x and y are on the right side of the equation.
What is the solution for |x - 5| ≤ 0?
{5} Absolute value of a number can never be < 0 but it can be 0 when x =5. |5 - 5| = 0
How do you find the x-intercept in a linear equation? ax + by = c
Set y = 0 to isolate the x.
What is the solution for |x - 5| ≥ 0?
R. Any real number, because the absolute value of a number is always ≥ 0.
How do we calculate slope from two given points? (x¹,y¹) (x²,y²)
y² - y¹/x² - x¹ to get the slope. (rise/run)
How should you decide what side will be shaded in while graphing a linear inequality? y ≥ 1/2x - 3
Test it by putting in a point, typically (0 , 0) and see if it's true. 0 ≥ 1/2(0) -3 is 0 ≥ 3, which means true. Shade on the side of the point. If false shade on opposite side.
Parallel lines:
Two lines with equal slopes are...
What do you do when you divide a negative number in an inequality?
Switch the signs
Describe this inequality in interval notation: x < 5/3 ?
( -∞ , 5/3)
The graph of x = b is?
A vertical line through b on the x axis. (Slope undefined)
How do you find the y-intercept in a linear equation? ax + by = c
Set x = 0 to isolate the y.
Perpendicular lines:
Two lines whose slopes are negative reciprocal of each other are...
How would you describe this inequality in interval notation: x ≥ 2 ?
[ 2 , ∞ )
A linear system where two lines are parallel (no answer) would be what type of system?
An inconsistent system.
Section 5
(50 cards)
What is the first step in graphing a linear inequality? y ≥ 1/2x - 3
Decide what type of line it will be. In this case it's a solid line because of the ≥ sign.
How is a positive line viewed?
It runs upwards. m>0
What do you do when you divide a negative number in an inequality?
Switch the signs
How would you describe this inequality in interval notation: x ≥ 2 ?
[ 2 , ∞ )
A linear system where two lines cross (one answer) would be what type of system?
A consistent system.
When solving a linear system of equations, if the substitution or addition method resulted in 2 ≠ 4, what system would you have?
An inconsistent system, parallel lines. No solution.
What is the formula for slope-intercept form?
y = mx + b
A linear system where two lines are parallel (no answer) would be what type of system?
An inconsistent system.
What is the equation of the line through (0 , 2) that is perpendicular to the graph of the line: y = 1/2x - 4 ? (Think of the perpendicular rule)
y = -2/1x + 2 because (0 , 2) gives us the y intercept and -2/1 is the reciprocal of the line in the question.
What is the standard form for a linear equation?
ax + by = c Both x and y are on the right side of the equation.
How would you describe this inequality in a solution set: x < 5 ?
{x|x<5}
How do you find the x-intercept in a linear equation? ax + by = c
Set y = 0 to isolate the x.
Is this a linear equation or a linear function: f(x) = 2x - 1 ?
A linear function
What is the point slope equation? Used when you have a slope and a random point.
y - y¹ = m(x - x¹)
How would you word the function (front) part of this equation? P(x) = 7.25x
P is a function of x
How would you express this inequality in a solution set: -4 < t ≤ 5/3 ?
{t| -4 < t ≤ 5/3}
What is the first step in this equation : 2|2x+1| -4 = 16 ?
Get rid of the variables outside of the absolute value.
What will the solution of this absolute value equation look like |exp.| < a ?
-a < exp < a
A linear system where two lines are directly on top of each other (infinite answers) would be what type of system?
A dependent system.
How is slope defined?
The slope of a line measures the tilt of the line.
How should you decide what side will be shaded in while graphing a linear inequality? y ≥ 1/2x - 3
Test it by putting in a point, typically (0 , 0) and see if it's true. 0 ≥ 1/2(0) -3 is 0 ≥ 3, which means true. Shade on the side of the point. If false shade on opposite side.
These two equations would be easiest to solve by what method Equation¹: y = -2x + 6 Equation²: 3x - 2y = 16?
The substitution method, substitue first equation in place of y in the second: 3x - 2(-2x + 6) = 16 and solve.
The graph of x = b is?
A vertical line through b on the x axis. (Slope undefined)
These two equations would be easiest to solve by what method Equation¹: 4x + 2y = 6 Equation²: 3x - 2y = 16?
The addition method. If set up as if adding them together the y's would cancel. Then you could solve for x and then solve for y.
What will the solution of this absolute value equation look like |exp.| = a ?
exp = a or exp = -a
When graphing linear inequalities, for what symbols would you use a solid line in the graph?
≤ and ≥
Describe this inequality in interval notation: x < 5/3 ?
( -∞ , 5/3)
What is the solution for |x - 5| < 0?
∅ The absolute value of a number can never be < 0. No solution.
What will the solution of this absolute value equation look like |exp.| > a ?
exp. < -a or exp. > a
When solving a linear system of equations, if the substitution or addition method resulted in 2 = 2, what would you have?
A dependent system, two lines on top of each other. ∞ Solutions
How is a negative line viewed?
It runs downwards. m<0
What is the solution for |x - 5| ≤ 0?
{5} Absolute value of a number can never be < 0 but it can be 0 when x =5. |5 - 5| = 0
Is this a linear equation or a linear function: y = 2x - 1 ?
A linear equation.
What three methods could be used to solve systems of linear equations? Equation¹: 2x + y = 6 Equation²: 3x - 2y = 16
By graphing, the substitution method, and the addition method.
For what inequality symbols do you use parenthesis?
< and >
If m=0 what type of line would be viewed on the graph?
A horizontal line.
For what inequality symbols do you use brackets?
≤ and ≥
How do we view lines on the graph to get the correct slope?
Left to right
What is the solution for |x - 5| ≥ 0?
R. Any real number, because the absolute value of a number is always ≥ 0.
How do we calculate slope from two given points? (x¹,y¹) (x²,y²)
y² - y¹/x² - x¹ to get the slope. (rise/run)
How do you find the equation of a line if you are given two points, but no y-intercept or slope? Such as: (4 , 3) (6 , -2)
You would have to find the slope first, x² - x¹/y² - y¹, and then use the point slope equation, y - y¹ = m(x - x¹)
How does slope tell us the tilt? y/x
Because y/x = rise/run
How do you find the y-intercept in a linear equation? ax + by = c
Set x = 0 to isolate the y.
What do you do when you have a negative denominator? ex b-2/-5
Move it up top. answer: 2-b/5 (Make sure to change signs)
What is the difference in slopes for perpendicular lines?
They have negative reciprocals. ex. Line¹: m= 2/4, would mean Line²: m= -4/2
A linear system where two lines are directly on top of each other (infinite answers) would be what type of system?
A dependent system.
The graph of y = b is?
A horizontal line through b on the y axis. (Slope = 0)
What do parallel lines have in common?
Equal slopes. m¹ = m²
When graphing linear inequalities, for what symbols would you use a dotted line in the graph?
< and >
How do you graph the line of a linear inequality? (Also the 2nd step) y ≥ 1/2x - 3
Set the inequality as an equation y ≥ 1/2x - 3 turns into y = 1/2x - 3, and plot the line.
Section 6
(50 cards)
Example: 0/a
0.
With Verticle Lines...
The Slope is undefined!
Example: a/0
Not defined.
Real Numbers
2, -10, -131.3337, 1/3, etc. Real Numbers can be represented by decimal numbers. Real numbers include both the rational and irrational numbers.
What is Median?
The median (middle) value of a range of values.
What is the Domain?
x.
Increasing & Decreasing Functions
f increases on l if, whenever x^1<x^2, f(x^1)<f(x^2). f decreases on l if, whenever x^1<x^2, f(x^1)>f(x^2).
When solving a linear system of equations, if the substitution or addition method resulted in 2 ≠ 4, what system would you have?
An inconsistent system, parallel lines. No solution.
Classify: 13/7
Rational Number
Integers
-3, -2, -1, 0, 1, 2, 3, etc. These are the natural numbers, their additive inverses (negatives), and 0.
What is the Standard Form?
Natural Numbers (Counting Numbers)
1, 2, 3, 4, 5, 6, 7, 8, 9, etc. Whole numbers that are not negative.
What is the Difference Quotient?
How do we read? And how should I read a graph?
From left to right. Read a graph the same way.
What is an Ordered Pair?
(x, y)
When solving a linear system of equations, if the substitution or addition method resulted in 2 = 2, what would you have?
A dependent system, two lines on top of each other. ∞ Solutions
What is the Distance Formula?
What is a Linear Equation?
A Linear Equation in one variable is an equation that can be written in the form ax+b=0. It must be equal to 1 (variable). It can not be squared or cubed unless they cancel out. Some may not even have variables, which it then is undefined.
What is the Standard Equation of a Circle?
What is a Function?
A Function is a relation in which every element in a first set (domain) is paired to a unique element on a second set (range).
What are the "Library of Functions?"
How do you Solve Application Problems?
Classify: -1.2
Rational Number.
What is the Vertical Line Test?
If every vertical line intersects a graph at no more than one point, then the graph represents a function.
What is a Circle?
A Circle is a collection/set of points that are on a fixed distance (radius) from a fixed point (center).
What must you ALWAYS be careful with?
Your signs!
Absolute Value is...
Absolute Value is the distance from zero; always positive.
What is a Cartesian (rectangular) coordinate plane?
What is the Domain of a Function?
The Domain of a Function is the set of values that makes the function well defined. If the function is not well defined at a point, then that point is not in the domain.
What is the Slope-Intercept Form of an Equation of a Line?
Also known as a Linear Function.
What is Mean?
The quotient of the sum of several quantities and their number; an average.
Order of Operations is...
PEMDAS Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction.
The General Form of the Equation of a Line is...
Classify: 5
Natural Number, Integer, Rational Number.
What is the Range?
y.
What is Function Notation?
f(x)=y
What is the Slope Formula?
What is the Point-Slope Form of an Equation of a Line?
What does Relation mean?
A relation is a set of ordered pairs.
What is an Equation?
An Equation is a statement that two mathematical expressions are equal.
What is the U (Union) Symbol?
Joins two pairs. You write this in between the inequality of the interval notation.
How do you do Intersection-of-Graph method on a Calculator?
Type in your results using Y=, Graph and then hit 2nd>Calc and Intersect to find the Intersection. Can also be done by hand.
What are the three types of Equations?
1. Contradiction - No solution. 2. Identity - There is a solution. 3. Conditional - Satisfied by some, but not by all values of the variable.
Rational Numbers
2/1, 1/3, (-1/4), 22/7, 0, 1.2, etc. Every integer is a rational number. Rational numbers can be expressed as decimals that either terminate (end) or repeat a sequence of digits.
What is the Midpoint Formula?
What is Interval Notation and how do you use it?
What is the Range of a Function?
The Range of a Function f(x) is the set {y=f(x)}.
What are the Properties of Equality?
What is the Average Rate of Change?
The Average Rate of Change of f from x^1 to x^2 is... y^2-y^1/x^2-x^1
Irrational Numbers
Irrational Numbers are numbers which are no rational numbers. They cannot be expressed as the ratio of two integers and has a decimal representation that does not terminate or repeat.
Section 7
(50 cards)
Independent Variable
variable that is changed in an experiment
Constant of Variation
the number k in equations of the form y=kx
Standerd Form
ax+by=c
x-intercept
The x-coordinate of the point where a line crosses the x-axis.
Scatter Plot
a graph with points plotted to show a possible relationship between two sets of data.
How do you make a Perfect Square?
Function
A relation which one element from the domain is paired with range
What is the Quadratic Function Equation?
The "a" in f(x) is called the Leading Coefficient.
What is an easier alternate formula to find the vertex?
Mapping Diagram
a way to show a relation that links elements of the domain with cooresponding elements of the range
Stretch
Multiplys all y values by the same factor greater than 1
What is the Absolute Value Function?
It is V-Shaped and is represented by y=|x|. It cannot be represented by single linear function.
How do you do a Quadratic Equation by Calculator?
Use Y^1 & Y^2 in your Calc. Hit Graph, and then use the Intersection.
What is Set Builder Notation?
How can you Graph Quadratic Functions?
By hand or by using a calculator. A calculator is a good tool to check your answer by hand!
A Projectile is...
... anything you can throw.
Domain
The set of x-coordinates of the set of points on a graph; the set of x-coordinates of a given set of ordered pairs. The value that is the input in a function or relation.
What is the Square Root Property?
y-intercept
the y-coordinate of the point where the line crosses the y-axis.
Direct Variation
y=kx
Range
The y-coordinates of the set of points on a graph. Also, the y-coordinates of a given set of ordered pairs.
What is the basic Absolute Value Equation?
You can also have a number inside, such as |x-3| and the equal sign can be changed to something such as < or >.
Trend Line
a line that approximates the relationship between the data sets of a scatter plot
Slope Intersept Form
the equation of a line in the form of y=mx+b, where m is the slope and b is the y intersept
Linear Inequality
an inequality in two variables whose graph is a region of the coordinate plane that is bounded by a line
Function Notation
an equation in the form of 'f(x)=' to show the output value of a function, f, for an input value x
What is a Parabola?
The graph f(x) is called a Parabola. Examples: f(x) = 2x^2 - x + 1 (Opens Upward because positive. a>0.) g(x) = -x^2 + 3x -5 (Opens Downward because negitive. a<0.) The highest (Absolute Max)/lowest (Absolute Min) point is called the Vertex.
Shrink
Reduces y values by a factor between 1-0
What is a Compound Inequality?
Make sure to isolate the variable in between. The solution set (in interval notation) of a compound inequality is always an interval of these: (a,b), [a,b], (a,b], [a,b).
Relation
A set of ordered pairs
Parent Function
The simplest function in a family; all functions in the family are transformations of it
Linear Equation
an equation whose graph is a straight line
What is the Vertex Form?
Parabola graph of f(x) = a (x-h)^2 + k with a not equal to 0; vertex (h,k). For the Vertex, h is always the opposite and k is what is displayed. This formula is known as the Standard Form of the Parabola.
What are the Inequality Symbols?
Dependent Variable
a factor that can change in an experiment in response to changes in the independent variable
Transformation
An operation that moves or changes a geometric figure in some way to produce a new figure
Vertical Line Test
a method to determine if a graph is a function or not
How do you find the Area of a Rectangle?
What is the Formula when dealing with Distance, Speed, Time, etc.?
Absolute Value Function
a function written in the form y = /x/, and the graph is always in the shape of a v
Translation
A transformation that "slides" each point of a figure the same distance in the same direction.
Reflection
A transformation that "flips" a figure over a mirror or reflection line.
What can a Linear Function Equation be written as?
f(x) = ax + b or f(x) = mx + b. The formula for a quadratic function is different from that of a linear function because it contains an x^2 term. Examples: f(x) = 3x^2 + 3x + 5 and g(x) = 5 - x^2 are not linear equations because of the ^2's.
Slope
the steepness of a line, equal to the ratio of a vertical change to the corresponding horizontal change
Parameter
a determining or characteristic element; a factor that shapes the total outcome; a limit, boundary
Point slope form
Y-Y1=m(x-x1)
What is Completing the Square?
As a Quadratic Expression: x^2 + kx + (k/2)2. As a Perfect Square Trinomial: x^2 + kx + (k/2)^2 = (x+k/2)^2.
Completing the Square Example...
x^2 + 4x + 5 1. x^2 + 4x ____ + 5 2. 4/2. ALWAYS divide by 2! Then square the answer; 2^2 = 4. Now add AND THEN ALSO subtract it. 3. (x^2 + 4x + 4) + 5 - 4 4. Now make it a perfect square! (x + 2)^2 + 1. This is the Standard Form - f(x) = a(x - h)^2 + k. 1(x+2)^2+1; 1 is a, 2 is h and 1 is k. Vertex is (-2,1) and it is going Upwards.
Linear Function
a function in which the graph of the solutions forms a line
Vertex
The point in a function where the function reaches a min or a max
Section 8
(50 cards)
Complete Factored Form...
f(x) = a^n (x-c)(x-b)(x-n)... Such as... 7x^3 - 21x^2 + 7x + 21 = 7(x+1)(x-1)(x-3)
A Function is Neither if...
A Function is "Neither" if it is not odd nor even. Such as... + + +.
What types of Transformations are there?
There are many types of Transformations. All moving either left, right, up or down. Some are... y = f(x) + c --> Upward. y = f(x) - c --> Downward. y = f(x - c) --> Right. y = f(x + c) --> Left. If a number is in front of the X, this number makes it go slower (Ex. 1/2) or faster (Ex. 2).
Local/Relative Maximum (Minimum) & Absolute/Global Maximum (Minimum) is...
What is the Quadratic Formula?
Combining Transformations...
Ex. y = -2(x - 1)^2 + 3 - Reflects across the x-axis. 2 Stretches vertically by a factor of 2. - 1 Shifts to the right 1 unit. + 3 Shifts upward 3 units.
When doing Quadratic Equations...
When doing Quadratic Equations, you can separate the denominator. Such as... (-3/2) + or - (sqrt3/2) or (-3 + or - sqrt3)/(2)
Two ways to check when Dividing by Polynomials...
Dividend/Divisor = Quotient + Remainder/Divisor or Dividend = (Divisor)(Quotient) + (Remainder)
Two important things to remember in math are...
PEMDAS and to be careful with your signs!
Commutative Property
Change the order without changing the outcome.
Factoring can be done by...
... trial and error.
When dealing with shifting of graphs (ex.)...
Ex. from Exam 2... 1. f(x) = x^2 + 2x - 7; left 8 unit, up 11 units. 2. f(x+8) + 11 3. f(x+8) = (x+8)^2 + 2(x+8) - 7 + 11. Clean up and done!
The Fundamentals Theorem of Algebra is...
The Fundamentals Theorem of Algebra is a Polynomail of degree n with complex coefficients has a complex zero. Fact: A Polynomial of degree n has n zeros (counting multiplicity.)
What are Complex Numbers?
A Complex Number is a number than can be written as a + bi. A is Real and B is Imaginary. >0 (is postive) - 2 distinct real sol. >| (bottom line) ( is zero) - Two solutons, but same. So 1. <0 (is negative) - No real solutions.
A Function is Odd if...
f(-x) = -f(x) (- + -) Always Symmetrical with respect to the origin.
Synthetic Division...
A shortcut that can be used to divide h-k into a polynomial. Make sure that everything is written in decreasing order of powers!
The Factor Theroem...
In the Factor Theroem, X-K is a factor of P(X) if and only if the remainder is P(K) = 0. (The remainder is 0). Ex. Decide if x-3 is a factor of x^3 - 2x +1 Use Synthetic, Linear or Remainder Theorem. P(X) = x^3 - 2x + 1 P(3) = 3^3 - 2(3) + 1 = 22 It's not 0, so x-3 is not a factor.
Property
The actions of numbers when combined.
What is a Complex Conjugate?
If a + bi is given, then the Complex Conjugate is a - bi. Fact: (a + bi)(a - bi) = a^2 + b^2. The i disappears. Fact: (Conjugate Zeros Therom) - If a + bi is a solution at some polynomial eqn., then a - bi is also a solution.
The Discriminate Formula is...
Two rules when dealing with reflections and negatives are...
1. When negative is outside, it will make it look down. (Also the opposite.) 2. When negative is inside, it will make it look left. (Also the opposite.) * When dealing with square roots, when the negative is inside, the only values we can then add are those that are negative... which then make it a positive and is valid.
Associative Property
Change the grouping without changing the outcome.
Addition Property of Zero
Adding zero to a number is equal to the same number.
Base
Number getting multiplied by itself.
Power
Indicates the number of times a number is multiplied by itself.
Brackets
used to group things to be done first, or show multiplication.
What is a Transformation?
A Transformation is a shift or translations in the xy-plane.
Parentheses
Used to group things to be done first, or show multiplication.
Long Division...
Make sure that everything is written in decreasing order of powers!
When Dividing Polynomials, you must make sure...
Make sure that everything is written in decreasing order of powers! Ex. 3x^3 - x^2 + 5 --> 3x^3 - x^2 + 0x + 5
Order of Operations
Order to solve a problem. PEMDAS.
Conjugate Zeros Theorem is...
The Conjugate Zeros Theorem is if a Polynomial f(x) has only real coefficients and if a + bi is a zero of f(x), then the conjugate a - bi is also a zero f(x). Ex. f(x) = 1 + 2i is a zero. f(x) = 1 - 2i is also a zero.
Exponent
Indicates how many times the base gets multiplied by itself.
Real Zero's of a Polynomial facts...
Real Zero's of a Polynomial = x-intercept. No x-intercept means no real zeros.
Rational Zero's Test is...
The possible Rational Zero's of f(x) are: factors of a^o / factors of a^n. Steps: 1. Consider the possible rational zeros. 2. Find the zero's of the quotient. 3. Write f(x) in factoral form.
These (dealing with i & Transformations) can be done by...
Hand and/or Calc.
Composite
A number that has three or more whole number factors.
Distributive Property
Distribute the the number to the other ones.
Factor
Number multiplied by another factor to get a product in a multiplication problem.
Additive Identity Property
Addition Propery of Zero
What does Area equal?
A Function is Even if...
f(-x) = f(x) (+ - +) Always Symmetrical with respect to the y-axis.
Basics of division...
What is the Zero-Product Property?
ab=0 if and only if a=0 or b=0.
What is an Imaginary Unit?
There is also more, but the most commonly used is the one above and i^2 = -1.
The Remainder Theroem...
The Remainder Theroem is when the remainder of dividing P(X) by X-K is P(K). You can also use Linear or Synthetic of course. Ex. f(2) = (2)^4 - 3(2)^2 - 4(2)^2 + 12(2)... Note: P(X) = (X-K) Q(X) + R If the remainder is 0, P(X) = (X-K) Q(X). Thus, (X-K) (The divisor) is a factor of P(X).
Prime
A number that has no other factors but itself and 1.
What is a Polynomial Function?
For example, 2x^2 - 3x + 1 is a Polynomial; C(X) = 4 is a Polynomial... you just don't see the "0's."
What are some Complex Numbers?
1 + 2i, -3i, 4, 3 - 11i, etc. Every Real Number is a Complex Number (Ex. 4 --> 4+0i).
What is the Expression of sqrt(-a)?
If a > 0, then sqrt(-a) = i sqrt(a).
Section 9
(50 cards)
What is the Range of a Function?
The Range of a Function f(x) is the set {y=f(x)}.
What is Mean?
The quotient of the sum of several quantities and their number; an average.
Universal Set
All the elements you can use.
What is the Distance Formula?
Complement
Everything that's NOT in that set.
Origin
Intersection of the two axes.
What is the Midpoint Formula?
Coordinate Plane
Plane determined by the axes
Graph of the Ordered Pair
Graph that order pairs get plotted onto.
Intersection
All the elements that belong to both sets.
Coordinate axes
lines that have the same scale and are drawn perpendicular to eachother.
Absolute Value is...
Absolute Value is the distance from zero; always positive.
X- Axis
Horizontal line
What does Relation mean?
A relation is a set of ordered pairs.
Classify: -1.2
Rational Number.
Classify: 13/7
Rational Number
Y- Axis
Vertical line
Irrational Numbers
Irrational Numbers are numbers which are no rational numbers. They cannot be expressed as the ratio of two integers and has a decimal representation that does not terminate or repeat.
Example: 0/a
0.
x- coordinate
(Abscissa) 1st number in the pair
Integers
-3, -2, -1, 0, 1, 2, 3, etc. These are the natural numbers, their additive inverses (negatives), and 0.
Polygon
Closed figure whose sides are line segments.
What is an Ordered Pair?
(x, y)
What is the Range?
y.
What is a Cartesian (rectangular) coordinate plane?
What is the Domain of a Function?
The Domain of a Function is the set of values that makes the function well defined. If the function is not well defined at a point, then that point is not in the domain.
Example: a/0
Not defined.
What is Median?
The median (middle) value of a range of values.
Empty Sets
Disjoint Sets { } or null.
What must you ALWAYS be careful with?
Your signs!
Order of Operations is...
PEMDAS Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction.
Multiplication Property of Zero
Multiplying 0 by a number is equal to 0.
Multiplication Property of 1
Multiplying 1 by a number is equal to the same number.
What is the Domain?
x.
Natural Numbers (Counting Numbers)
1, 2, 3, 4, 5, 6, 7, 8, 9, etc. Whole numbers that are not negative.
Quadrants
4 regions, roman numerals 1, 2, 3 and 4, in a counterclockwise order.
Classify: 5
Natural Number, Integer, Rational Number.
Rational Numbers
2/1, 1/3, (-1/4), 22/7, 0, 1.2, etc. Every integer is a rational number. Rational numbers can be expressed as decimals that either terminate (end) or repeat a sequence of digits.
What is a Function?
A Function is a relation in which every element in a first set (domain) is paired to a unique element on a second set (range).
What is a Circle?
A Circle is a collection/set of points that are on a fixed distance (radius) from a fixed point (center).
Multiplicative Inverses (Reciprocals)
Multiplication Property of 1
Quadrilateral
polygon with 4 sides.
Verticies
Endpoints of the sides.
Real Numbers
2, -10, -131.3337, 1/3, etc. Real Numbers can be represented by decimal numbers. Real numbers include both the rational and irrational numbers.
What is Function Notation?
f(x)=y
Disjoint Sets
Empty Sets { } or null.
Union
Everything that belongs to BOTH sets.
What is the Standard Equation of a Circle?
y- coordinate
(Ordinate) 2nd number in the pair.
Ordered Pair
two numbers to be plotted.
Section 10
(50 cards)
What is the Average Rate of Change?
The Average Rate of Change of f from x^1 to x^2 is... y^2-y^1/x^2-x^1
Slope
Compound Interest Formula solved for P
How do you Solve Application Problems?
Quadratic Equation
What is the Vertical Line Test?
If every vertical line intersects a graph at no more than one point, then the graph represents a function.
Permutations Formula
What is Set Builder Notation?
Midpoint Formula
What is a Compound Inequality?
Make sure to isolate the variable in between. The solution set (in interval notation) of a compound inequality is always an interval of these: (a,b), [a,b], (a,b], [a,b).
Slope Intercept
Combinations Formula
What are the "Library of Functions?"
What is the Formula when dealing with Distance, Speed, Time, etc.?
Log Power Rule
logαMⁿ =nlogαM
P(n,r) = n!/(n-r)!
Permutations Formula
Increasing & Decreasing Functions
f increases on l if, whenever x^1<x^2, f(x^1)<f(x^2). f decreases on l if, whenever x^1<x^2, f(x^1)>f(x^2).
How do we read? And how should I read a graph?
From left to right. Read a graph the same way.
What is the Slope-Intercept Form of an Equation of a Line?
Also known as a Linear Function.
Stnd Form of a Circle
What are the Properties of Equality?
Distance
What is Interval Notation and how do you use it?
Continuously Compound Interest Formula
What is the basic Absolute Value Equation?
You can also have a number inside, such as |x-3| and the equal sign can be changed to something such as < or >.
With Verticle Lines...
The Slope is undefined!
What is the U (Union) Symbol?
Joins two pairs. You write this in between the inequality of the interval notation.
Vertex Formula
Radioactive Decay Formula
Af=Ai(2)-t/h Where Af is Amount Final(at present) is equal to Amount Initial times two raised to the negative of the amount of time passed divided by the Halflife.
What is the Absolute Value Function?
It is V-Shaped and is represented by y=|x|. It cannot be represented by single linear function.
What is a Linear Equation?
A Linear Equation in one variable is an equation that can be written in the form ax+b=0. It must be equal to 1 (variable). It can not be squared or cubed unless they cancel out. Some may not even have variables, which it then is undefined.
What is an Equation?
An Equation is a statement that two mathematical expressions are equal.
nCr = n!/r!(n-r)!
Combinations Formula
How do you do Intersection-of-Graph method on a Calculator?
Type in your results using Y=, Graph and then hit 2nd>Calc and Intersect to find the Intersection. Can also be done by hand.
General Form of a Circle
Compound Interest Formula
Pascals Triangle
What is the Difference Quotient?
Testing Symmetry
Logarithmic Relationships
Binomial Expansion
What is the Standard Form?
What are the three types of Equations?
1. Contradiction - No solution. 2. Identity - There is a solution. 3. Conditional - Satisfied by some, but not by all values of the variable.
Difference Quotient
Log Change of Base Formula
What is the Slope Formula?
What are the Inequality Symbols?
The General Form of the Equation of a Line is...
Malthusian Population Growth
P = Pµeⁿ° Where P is current population, Pµ is initial population, e is raised to the ()number in years times the °growth rate (birth rate - death rate)
What is the Point-Slope Form of an Equation of a Line?
Section 11
(52 cards)
reciprocal
one of a pair of numbers whose product is 1: the reciprocal of 2/3 is 3/2
distributive property
a(b + c) = ab + ac an + ac = a(b+ c)
multiplication identity property
The product of any number and one is that number.
variable expression
consists of numbers, variables, and operations
terms
in an expression are separated by addition and subtraction signs
Point Slope Form
ratio
a comparison of two numbers by division
Inequality
A statement that compares two quantities using <, >, ≤,≥, or ≠
ordered pair
A pair of numbers, (x, y), that indicate the position of a point on a coordinate plane.
like terms
terms that have identical variable parts raised to the same power
area
the number of square units needed to cover a flat surface
equation
a mathematical sentence with an equal sign that shows that two expressions are equivalent
Summation Notation
solving an equation
finding all the solutions of an equation
rational number
A number that can be written as a/b where a and b are integers, but b is not equal to 0.
order of operations
the order in which operations in an expression to be evaluated are carried out. 1. parentheses 2. exponets 3. multiplication and divison 4. addition and subtraction
domain
the set of all the input (x-values) for a function
integers
Positive and Negative Whole Numbers and 0. Examples: .... -3, -2, -1, 0, 1, 2, 3.....(Does not include fractions or decimals)
additive inverse property
The sum of a number and its opposite is zero.
output
the y-value in a function
rate
a ratio that compares two quantities measured in different units
associative property of multiplication
changing the grouping of factors will not change the product, (ab)c = a(bc)
Solve for a variable
To get a variable alone on one side of an equation or inequality in order to solve the equation or inequality
perimeter
The sum of the lengths of the sides of a polygon
Binomial Expansion By Pascals Triangle
additive identity property
The sum of a number and zero is always that number.
Definition of Term in Binomial Expansion
range
the set of all the output (y-values) for a function
variable
a letter used to represent one or more numbers
constant
a term that has no variable and does not change
coefficient
number in front of a variable
Factorial Notation
n! = n × (n - 1) × (n - 2) × (n - 3) × ... × 3 × 2 × 1
base
is the number that is repeatedly multiplied in a power
numerical expression
consists of numbers and operations
commutative property of addition
In a sum, you can add terms in any order, a + b = b + a
exponent
a mathematical notation indicating the number of times a quantity is multiplied by itself
Equivalent Equations
equations that have the same solution
linear equation
an equation whose answers are ordered pairs (x,y) and whose answers graph a straight line
irrational number
a number that can not be written a/b
power
is a number made of repeated factors
coordinate plane
A plane that is divided into four regions by a horizontal line called the x-axis and a vertical line called the y-axis. aka "the Cartesian plane" after René Descartes
associative property of addition
changing the grouping of terms will not change the sum, (a + b) + c = a + (b + c)
Inverse Operation
operations that undo each other, such as addition and subtraction
quadrant
one of four sections into which the coordinate plane is divided
commutative property of multiplication
the order of the factors does not change the product a x b = b x a
function
a relation that assigns exactly one output value for each input value
evaluate
to find the value of an expression
input
the x-value in a function
x-intercept
the point where a graph crosses the x-axis
Standard Form of a Line
Section 12
(53 cards)
promoting
calling attention to, advertising or publicizing
in terms of
with regard to,with respect to
slope
the steepness of a line on a graph, rise over run
Range
the second term in an ordered pair
represents
under or below
reasonable conclusion
a position or opinion or judgment reached after considering facts and observations
effect
an outward appearance
approximately
Almost, but not exactly; more or less
per
by or through
non-included
Not between
shown
to reveal
Slope Intercept Form
y= mx + b where "m=slope" and "b=y-intercept"
composed of
made up of
view
the act of looking or seeing or observing
prove
prove formally
additional
existing or coming by way of addition
determine
shape or influence
chemical compound
a substance formed by the chemical combination of two or more elements in definite proportions
Absolute Value
a number's distance from zero
region
the approximate amount of something usually used prepositionally as in 'in the region of'
international
concerning or belonging to all or at least two or more nations
Domain
the first term in an ordered pair
Rational Numbers
numbers that stop and can be written as fractions
closets to
something that is close to another object
identical
exactly alike
common
to be expected
Coefficient
the number in front of the variable (ie: 5 in 5a)
joined
joined
regarding
concerning about or with respect to
elapsed
passed
conclusion
the act of making up your mind about something
Irrational Numbers
numbers that do not repeat or terminate
Slope Formula
Slope=m= Y2 - Y1 / X2 - X1
population
statistics the entire aggregation of items from which samples can be drawn
satisfy
fill or meet a want or need
routine
found in the ordinary course of events
corresponding
Angles or lines of 2 different polygons that are in the same position
best represents
closest to; most similar to
annual
occurring or payable every year
exceeds
to go beyond
sufficent
all that is needed, enough
systems
groups of organs working together to perform complex functions
formed
clearly defined
y-intercept
the point where a graph crosses the y-axis
counterexample
refutation by example
twice
two times
category
a classification or grouping
creates
to make, form.
shown
Revealed
regardless
in spite of everything
Section 13
(52 cards)
Monomial
a number, variable, or a product of a number and variable
Elimination Method
eliminate a variable by adding or subtracting the equations
Translation
movement of a figure
Relative Minimum
lowest point on a graph
Scalar Multiplication
multiplying a matrix by a scalar
Vertex
the point of intersection between lines
conclusion
the act of making up your mind about something
Determinant
numbers in a square array enclosed by parallel lines
Empty Set
when an equation has no solution
counterexample
refutation by example
Feasible Region
the area between intersecting lines (shaded area)
composed of
made up of
elapsed
passed
Synthetic Substitution
the use of synthetic division to evaluate a polynomial
formed
clearly defined
Function
one domain paired with exactly one range
creates
to make, form.
effect
an outward appearance
Independent Variable
the variable that makes up the domain
Discriminant
b²-4ac
Quadratic Equation
x=(-b±√b²-4ac)/2a
Relative Maximum
highest point on a graph
Leading Coefficient
the coefficient of the term with the highest degree
Scatter Plot
a graph of ordered pairs
Matrix
a rectangular array of numbers inside square brackets
Standard Form
Ax+By=c
annual
occurring or payable every year
corresponding
Angles or lines of 2 different polygons that are in the same position
common
to be expected
Dilation
enlargement or reduction of an image
Imaginary Unit
i = √-1
chemical compound
a substance formed by the chemical combination of two or more elements in definite proportions
identical
exactly alike
Synthetic Division
method used to divide polynomials by monomials
Polynomial Function
function represented by a monomial or a sum of monomials
Quadratic Function
an equation of the form x²
Systems of equations
2 or more equations with the same variable
Degree of a Polynomial
greatest degree of any term
approximately
Almost, but not exactly; more or less
Parabola
a "u" shaped graph y=x²
exceeds
to go beyond
Substitution Method
solve an equation and substitute it into another equation
Element
each number in a matrix
category
a classification or grouping
x-intercept
where a graph crosses the x-axis
additional
existing or coming by way of addition
Complex Number
a number in the form a+bi
Slope
change in y over the change in x (rise/run)
determine
shape or influence
Root
solutions to a quadratic equation
Section 14
(50 cards)
shown
Revealed
sufficent
all that is needed, enough
international
concerning or belonging to all or at least two or more nations
variable
a letter used to represent one or more numbers
population
statistics the entire aggregation of items from which samples can be drawn
region
the approximate amount of something usually used prepositionally as in 'in the region of'
in terms of
with regard to,with respect to
commutative property of multiplication
the order of the factors does not change the product a x b = b x a
terms
in an expression are separated by addition and subtraction signs
coordinate plane
A plane that is divided into four regions by a horizontal line called the x-axis and a vertical line called the y-axis. aka "the Cartesian plane" after René Descartes
associative property of multiplication
changing the grouping of factors will not change the product, (ab)c = a(bc)
regarding
concerning about or with respect to
systems
groups of organs working together to perform complex functions
integers
Positive and Negative Whole Numbers and 0. Examples: .... -3, -2, -1, 0, 1, 2, 3.....(Does not include fractions or decimals)
view
the act of looking or seeing or observing
best represents
closest to; most similar to
base
is the number that is repeatedly multiplied in a power
like terms
terms that have identical variable parts raised to the same power
area
the number of square units needed to cover a flat surface
variable expression
consists of numbers, variables, and operations
numerical expression
consists of numbers and operations
additive identity property
The sum of a number and zero is always that number.
order of operations
the order in which operations in an expression to be evaluated are carried out. 1. parentheses 2. exponets 3. multiplication and divison 4. addition and subtraction
power
is a number made of repeated factors
represents
under or below
distributive property
a(b + c) = ab + ac an + ac = a(b+ c)
closets to
something that is close to another object
perimeter
The sum of the lengths of the sides of a polygon
non-included
Not between
additive inverse property
The sum of a number and its opposite is zero.
constant
a term that has no variable and does not change
commutative property of addition
In a sum, you can add terms in any order, a + b = b + a
joined
joined
satisfy
fill or meet a want or need
routine
found in the ordinary course of events
coefficient
number in front of a variable
associative property of addition
changing the grouping of terms will not change the sum, (a + b) + c = a + (b + c)
equation
a mathematical sentence with an equal sign that shows that two expressions are equivalent
per
by or through
reasonable conclusion
a position or opinion or judgment reached after considering facts and observations
regardless
in spite of everything
shown
to reveal
solving an equation
finding all the solutions of an equation
prove
prove formally
promoting
calling attention to, advertising or publicizing
twice
two times
evaluate
to find the value of an expression
exponent
a mathematical notation indicating the number of times a quantity is multiplied by itself
multiplication identity property
The product of any number and one is that number.
ordered pair
A pair of numbers, (x, y), that indicate the position of a point on a coordinate plane.
Section 15
(52 cards)
domain
the set of all the input (x-values) for a function
5 ∙ 0 = 0
n ∙ 0 = 0
5 + 4 = 4 + 5
n + m = m + n
6 + 0 = 6
n + 0 = n
7/8 ∙ 1 = 7/8
n ∙ 1 = n
¼ - ¼ = 0
n - n = 0
6 + 1.3 = 1.3 + .6
n + m = m + n
Equivalent Equations
equations that have the same solution
x-intercept
the point where a graph crosses the x-axis
2/5 + ¼ = ¼ + 2/5
n + m = m + n
(.2)(0) = 0
n ∙ 0 = 0
34 - 34 = 0
n - n = 0
(-5)(2) = (2)(-5)
nm = mn
2/3 + 0 = 2/3
n + 0 = n
y-intercept
the point where a graph crosses the y-axis
Slope Intercept Form
y= mx + b where "m=slope" and "b=y-intercept"
9 ∙ 1 = 9
n ∙ 1 = n
rate
a ratio that compares two quantities measured in different units
(1)(1) = 1
n ∙ 1 = n
rational number
A number that can be written as a/b where a and b are integers, but b is not equal to 0.
.25 + 0 = .25
n + 0 = n
(-5)1 = -5
nm = mn
ratio
a comparison of two numbers by division
slope
the steepness of a line on a graph, rise over run
linear equation
an equation whose answers are ordered pairs (x,y) and whose answers graph a straight line
(-6)(-6) = (6)2
n ∙ n = n[sq]
5 - 5 = 0
n - n = 0
2 ∙ 2 = 22
n ∙ n = n[sq]
input
the x-value in a function
Solve for a variable
To get a variable alone on one side of an equation or inequality in order to solve the equation or inequality
function
a relation that assigns exactly one output value for each input value
output
the y-value in a function
0 ∙ 1 = 0
n ∙ 0 = 0
irrational number
a number that can not be written a/b
(½)(½) = (½)2
n ∙ n = n[sq]
3 ∙ 3 = 32
n ∙ n = n[sq]
range
the set of all the output (y-values) for a function
(.6)(1) = 1
n ∙ 1 = n
4 ∙ 7 = 7 ∙ 4
nm = mn
(.9)(.9) = (.9)2
n ∙ n = n[sq]
72 + 0 = 72
n + 0 = n
Inequality
A statement that compares two quantities using <, >, ≤,≥, or ≠
reciprocal
one of a pair of numbers whose product is 1: the reciprocal of 2/3 is 3/2
0 ∙ 3 = 0
n ∙ 0 = 0
(¾)(0) = 0
n ∙ 0 = 0
(.4)(.6) = (.6)(.4)
nm = mn
Inverse Operation
operations that undo each other, such as addition and subtraction
Slope Formula
Slope=m= Y2 - Y1 / X2 - X1
9.4 - 9.4 = 0
n - n = 0
quadrant
one of four sections into which the coordinate plane is divided
Section 16
(51 cards)
complex number
z=a+bi where i=√-1
½ ∙ 2 = 1
n ∙ 1/n = 1
Orthonormal Basis
a subspace spanned by an orthogonal set of unit vectors
orthogonality
Subspaces and orthogonality: A vector is only in a space (that is transposed) if it is orthogonal to every other vector in that space. WT is a subspace of Rn
Indefinite
Q(x) assumes both positive and negative values Eigenvalues are both positive and negative
spectral theorem
if A is symmetric, then it has real eigenvalues and can be orthogonally diagonalized
Eigenspace
the set of all solutions for Ax=λx for a specific λ; the null-space of (A-λI)
Cauchy-Schwarz inequality
We use this to find the angle between two real vectors in terms of approximations; (find out more!)
unit vector
A vector with a length of one (unit, hence the name)
quadratic form
A function (Q()) where the vector inputed (x) = x^T Ax, where A is a symmetric matrix
Linear Substitution
find out about linear substitution
angles between real vectors
U·V=‖‖U‖‖ ‖‖V‖‖ cos (ϴ); ϴ=cos-1(U·V)/U·V
orthogonal diagonalization (of a real symmetric matrix)
When an orthogonal matrix P with a diagonalized matrix D where A=PDP⁻¹=PDP^T (P⁻¹=P^T)
Change of Variable
To change: 1. orthogonally diagonalize matrix A 2. Make D a diagonal matrix with eigenvalues 3.substitute x^T Ax with (Py)^T A (Py) where x = Py 4.simplify to get λy² +λy²...
7 ∙ 1/7 = 1
n ∙ 1/n = 1
.3/.3 = 1
n/n = 1
modulus
(length of a complex number) z=√(a²+b²)=‖a,b‖
Trace (of a square matrix)
Sum of the diagonal entries in a square matrix. Also important: It is the sum of the eigenvalues of A
Length (complex and real vectors)
The square root of all the entries of the vector squared, if it is a vector with complex entries than you must use the absolute values of the complex numbers.
orthogonal vector
a set of non-zero vectors; angle between vectors is 90; the dot product is zero
Principle Axes Theorem
Let A be a symmetric matrix; there is an orthogonal change of variable, x = Py, that transforms xTAx into yTDy with no cross-products
Orthogonal Projection
turning a vector into two other vectors r that sum up to it requires an orthogonal projection; Finding the weights for the linear combination: C1 =Y·U1/U1·U1 , given that y=c1v1+c2v2...
y = x²
Gram-Schmidt Process
used for making orthonormal/orthogonal bases; The process, which makes a set of vectors {x1..xn} into an orthogonal basis {v1...vn} 1. v1 = x1 2. v2 =x1- (x2·v1/v1·v1)v1 3. v3 = x3- (x3·v1/v1·v1)v1-x3·v2/v2·v2)v2 (etc.)
Algebraic Multiplicity
number of times an eigenvalue repeats. Or the number of times that a root appears in the characteristic polynomial.
Orthogonal Matrix
a square invertible matrix such that U-1=UT; has orthonormal columns and rows
y = x² − 3
Positive Semidefinite
Q(x)0 for all x Eigenvalues are all non-negative
9/9 = 1
n/n = 1
5/8 ÷ 5/8 = 1
n/n = 1
Eigenvalue
a scalar, nonzero solution to Ax=λx;Basically it is when you have a vector multiplied by a given matrix that produces another vector, which can be represented as a scalar (eigenvalue) and the original vector; Found by solving the characteristic polynomial (they are the roots)
Characteristic (polynomials of square matrices)
Found by setting det(A-I)=0, a scalar equation that gives us roots of eigenvalues
Spectrum (of a matrix)
the range or set of all eigenvalues of a square
Least Square fit
approximating inconsistent systems of Ax = b; the smaller difference of ||b-Ax||, the better the approximation; ATAx(approx)=ATb(approx)
conjugate (of complex numbers)
z=a-bi, etc.
Eigenvector
a nonzero vector (x) where Ax=λx;basically its a scaled vector, where the linear transformation doesn't change; It's also a nonzero in the nullspace of a given matrix [think(A-I)x=0
Transpose Properties (of a matrix)
1. (A^t)^t = A 2. (AB)^t = Bt At 3. (cA)t = c (A)t
7/8 ∙ 8/7 = 1
n ∙ 1/n = 1
3 ∙ 1/3 = 1
n ∙ 1/n = 1
spectral decomposition
A=1u1uT1 +2u2uT2....... where u are the columns of P in A=PDPTwhere PT=P-1; this is the equation for matrix A where the spectrum of eigenvalues determine each piece
Orthogonal Basis
Where all the vectors in the set are orthogonal to each other that forms a subspace; a.k.a where an orthogonal set that is also a subspace
symmetric matrix
A matrix such that it equals its transpose; any two of its eigevectors from different eigenspaces ( made from different eigenvalues) are orthogonal; counting multiplicities, has as many eigenvalues as rows or columns;orthogonally diagonizable.
similar matrix
When two matrices A, B and another (invertible) matrix P satisfy A=P⁻¹BP
Geometric Multiplicity
dimension of eigenspace (number of free variables)
Positive definite
Q(x)0 for all x≠0 All eigenvalues are all positive
Negative definite
Q(x)0 for all x≠0 All eigenvalues are all negative
y = x² + 3
diagonalizable matrix
the factorization of a matrix into three other matrices, PDP⁻¹, where D is a matrix containing the eigenvalues
14/14 = 1
n/n = 1
Orthonormal Vector
orthogonal vectors which have a length of 1(are unit vectors)
Section 17
(50 cards)
y = ± √(x − 5)
y = −x²
y = (x−1)² +3
b = 4
This graph does NOT represent ax² -1/4, where a would be a fraction less than 1. This graph is of the form (x⁴ - a)/(x² + b) Find the integer, b. HINT Type "b= ..."
additive identity property
The sum of a number and zero is always that number.
associative property of addition
changing the grouping of terms will not change the sum, (a + b) + c = a + (b + c)
y = −x² −2x
y = ±√x + 3
y = (x+3)² −5
base
is the number that is repeatedly multiplied in a power
additive inverse property
The sum of a number and its opposite is zero.
y = ±√(x+3)
y = ±√(−x+3)
y = −5x²
y = (x − 3)²
evaluate
to find the value of an expression
y = x² + 3x
y = (x+1)²
area
the number of square units needed to cover a flat surface
numerical expression
consists of numbers and operations
y = ±√x − 5
y = (x−2)²
y = (x−2)² +5
variable expression
consists of numbers, variables, and operations
y = x² + 10
perimeter
The sum of the lengths of the sides of a polygon
commutative property of addition
In a sum, you can add terms in any order, a + b = b + a
power
is a number made of repeated factors
y = (x+5)²
y = ±√−x
distributive property
a(b + c) = ab + ac an + ac = a(b+ c)
y = ±√x
variable
a letter used to represent one or more numbers
integers
Positive and Negative Whole Numbers and 0. Examples: .... -3, -2, -1, 0, 1, 2, 3.....(Does not include fractions or decimals)
associative property of multiplication
changing the grouping of factors will not change the product, (ab)c = a(bc)
exponent
a mathematical notation indicating the number of times a quantity is multiplied by itself
y = x² −25
multiplication identity property
The product of any number and one is that number.
order of operations
the order in which operations in an expression to be evaluated are carried out. 1. parentheses 2. exponets 3. multiplication and divison 4. addition and subtraction
y = ±√−x + 2
c) (x⁴−9)/(x²+1)
Which function is shown in blue? a) (x⁴−9)/(x²+9) b) (x⁴−9)/(x²+2) c) (x⁴−9)/(x²+1)
y = − (1/5)x²
commutative property of multiplication
the order of the factors does not change the product a x b = b x a
solving an equation
finding all the solutions of an equation
y = x² −5x
y = x² − 9
y = x² −3x
equation
a mathematical sentence with an equal sign that shows that two expressions are equivalent
y = (x+2)²
y = x² − 4
Section 18
(54 cards)
power
is a number made of repeated factors
reciprocal
one of a pair of numbers whose product is 1: the reciprocal of 2/3 is 3/2
exponent
a mathematical notation indicating the number of times a quantity is multiplied by itself
Solve for a variable
To get a variable alone on one side of an equation or inequality in order to solve the equation or inequality
ratio
a comparison of two numbers by division
x-intercept
the point where a graph crosses the x-axis
equation
a mathematical sentence with an equal sign that shows that two expressions are equivalent
numerical expression
consists of numbers and operations
integers
Positive and Negative Whole Numbers and 0. Examples: .... -3, -2, -1, 0, 1, 2, 3.....(Does not include fractions or decimals)
commutative property of multiplication
the order of the factors does not change the product a x b = b x a
Slope Formula
Slope=m= Y2 - Y1 / X2 - X1
function
a relation that assigns exactly one output value for each input value
area
the number of square units needed to cover a flat surface
constant
a term that has no variable and does not change
Equivalent Equations
equations that have the same solution
slope
the steepness of a line on a graph, rise over run
evaluate
to find the value of an expression
irrational number
a number that can not be written a/b
terms
in an expression are separated by addition and subtraction signs
terms
in an expression are separated by addition and subtraction signs
Slope Intercept Form
y= mx + b where "m=slope" and "b=y-intercept"
coordinate plane
A plane that is divided into four regions by a horizontal line called the x-axis and a vertical line called the y-axis. aka "the Cartesian plane" after René Descartes
y-intercept
the point where a graph crosses the y-axis
base
is the number that is repeatedly multiplied in a power
multiplication identity property
The product of any number and one is that number.
coefficient
number in front of a variable
perimeter
The sum of the lengths of the sides of a polygon
domain
the set of all the input (x-values) for a function
like terms
terms that have identical variable parts raised to the same power
ordered pair
A pair of numbers, (x, y), that indicate the position of a point on a coordinate plane.
rational number
A number that can be written as a/b where a and b are integers, but b is not equal to 0.
variable
a letter used to represent one or more numbers
Inverse Operation
operations that undo each other, such as addition and subtraction
associative property of multiplication
changing the grouping of factors will not change the product, (ab)c = a(bc)
commutative property of addition
In a sum, you can add terms in any order, a + b = b + a
solving an equation
finding all the solutions of an equation
quadrant
one of four sections into which the coordinate plane is divided
associative property of addition
changing the grouping of terms will not change the sum, (a + b) + c = a + (b + c)
Inequality
A statement that compares two quantities using <, >, ≤,≥, or ≠
like terms
terms that have identical variable parts raised to the same power
linear equation
an equation whose answers are ordered pairs (x,y) and whose answers graph a straight line
order of operations
the order in which operations in an expression to be evaluated are carried out. 1. parentheses 2. exponets 3. multiplication and divison 4. addition and subtraction
additive inverse property
The sum of a number and its opposite is zero.
range
the set of all the output (y-values) for a function
variable expression
consists of numbers, variables, and operations
input
the x-value in a function
additive identity property
The sum of a number and zero is always that number.
rate
a ratio that compares two quantities measured in different units
output
the y-value in a function
distributive property
a(b + c) = ab + ac an + ac = a(b+ c)
Section 19
(53 cards)
coefficient
number in front of a variable
Inverse Operation
operations that undo each other, such as addition and subtraction
y = x² − 3
slope
the steepness of a line on a graph, rise over run
y = −5x²
y = (x+1)²
constant
a term that has no variable and does not change
y = x² − 4
y = ±√(x+3)
y = −x²
y = ±√x + 3
Equivalent Equations
equations that have the same solution
y = x² + 3
Slope Formula
Slope=m= Y2 - Y1 / X2 - X1
y = x² + 3x
y = (x+3)² −5
function
a relation that assigns exactly one output value for each input value
y = (x+5)²
output
the y-value in a function
y = ± √(x − 5)
ordered pair
A pair of numbers, (x, y), that indicate the position of a point on a coordinate plane.
Solve for a variable
To get a variable alone on one side of an equation or inequality in order to solve the equation or inequality
x-intercept
the point where a graph crosses the x-axis
coordinate plane
A plane that is divided into four regions by a horizontal line called the x-axis and a vertical line called the y-axis. aka "the Cartesian plane" after René Descartes
y = x² + 10
rational number
A number that can be written as a/b where a and b are integers, but b is not equal to 0.
y = (x − 3)²
irrational number
a number that can not be written a/b
y = (x+2)²
y = (x−1)² +3
y = x² −3x
reciprocal
one of a pair of numbers whose product is 1: the reciprocal of 2/3 is 3/2
y = (x−2)²
quadrant
one of four sections into which the coordinate plane is divided
y = x² −25
ratio
a comparison of two numbers by division
Slope Intercept Form
y= mx + b where "m=slope" and "b=y-intercept"
y = ±√x − 5
y = x² − 9
y = x²
domain
the set of all the input (x-values) for a function
y-intercept
the point where a graph crosses the y-axis
linear equation
an equation whose answers are ordered pairs (x,y) and whose answers graph a straight line
input
the x-value in a function
range
the set of all the output (y-values) for a function
y = (x−2)² +5
rate
a ratio that compares two quantities measured in different units
Inequality
A statement that compares two quantities using <, >, ≤,≥, or ≠
y = ±√x
y = − (1/5)x²
Section 20
(53 cards)
y = ±√−x + 2
multiplication identity property
The product of any number and one is that number.
base
the bottom of a triangle
ordered pair
A pair of numbers, (x, y), that indicate the position of a point on a coordinate plane.
composite
a number that has more than two factors
associative property of addition
changing the grouping of terms will not change the sum, (a + b) + c = a + (b + c)
Isolate the variable
To get a variable alone on one side of an equation or inequality in order to solve the equation or inequality
b = 4
This graph does NOT represent ax² -1/4, where a would be a fraction less than 1. This graph is of the form (x⁴ - a)/(x² + b) Find the integer, b. HINT Type "b= ..."
equation
a mathematical sentence with an equal sign that shows that two expressions are equivalent
exponent
a mathematical notation indicating the number of times a quantity is multiplied by itself
mean
the sum of the values in a data set divided by the number of values in the set
evaluate
to find the value of an expression
area
the number of square units needed to cover a flat surface
solving an equation
finding all the solutions of an equation
power
is a number made of repeated factors
y = x² −5x
integers
Positive and Negative Whole Numbers and 0. Examples: .... -3, -2, -1, 0, 1, 2, 3.....(Does not include fractions or decimals)
prime number
A whole number that has exactly two factors, 1 and itself.
coefficient
number in front of a variable
absolute value
the distance a number is from 0 on the number line, value of n is written l n l
formula
An equation that shows a relationship among two or more quantities
distributive property
a(b + c) = ab + ac an + ac = a(b+ c)
commutative property of multiplication
the order of the factors does not change the product a x b = b x a
perimeter
The sum of the lengths of the sides of a polygon
terms
in an expression are separated by addition and subtraction signs
prime factorization
a number written as the product of its prime factors
coordinate plane
A plane that is divided into four regions by a horizontal line called the x-axis and a vertical line called the y-axis. aka "the Cartesian plane" after René Descartes
Equivalent Inequalities
inequalities that have the same solution
variable expression
consists of numbers, variables, and operations
Inverse Operation
operations that undo each other, such as addition and subtraction
additive identity property
The sum of a number and zero is always that number.
like terms
terms that have identical variable parts raised to the same power
monomial
a single term made up of numbers and variables
order of operations
the order in which operations in an expression to be evaluated are carried out. 1. parentheses 2. exponets 3. multiplication and divison 4. addition and subtraction
quadrant
one of four sections into which the coordinate plane is divided
numerical expression
consists of numbers and operations
commutative property of addition
In a sum, you can add terms in any order, a + b = b + a
y = −x² −2x
greatest common factor
The largest factor that two or more numbers have in common.
associative property of multiplication
changing the grouping of factors will not change the product, (ab)c = a(bc)
constant
a term that has no variable and does not change
y = ±√(−x+3)
c) (x⁴−9)/(x²+1)
Which function is shown in blue? a) (x⁴−9)/(x²+9) b) (x⁴−9)/(x²+2) c) (x⁴−9)/(x²+1)
variable
a letter used to represent one or more numbers
base
is the number that is repeatedly multiplied in a power
additive inverse property
The sum of a number and its opposite is zero.
y = ±√−x
height
the perpendicular (90 degrees) distance from the base of a triangle to the opposite vertex
Equivalent Equations
equations that have the same solution
Inequality
A statement that compares two quantities using <, >, ≤,≥, or ≠
Section 21
(53 cards)
input
the x-value in a function
opposite
the number that is on the other side of 0 and is exactly the same distance away from 0
range
is the difference of the greatest value and the least value in a set of data.
scale model
a model of an object in which the dimensions are in proportion to the actual dimensions of the object.
radius
a line segment from the center of a circle to any point on the circle (is also half the diameter)
x-intercept
the point where a graph crosses the x-axis
cube
special polyhedron with faces that are all squares
simplest form
when the GCF of the numerator and denominator is 1
percent
a ratio whose denominator is 100
output
the y-value in a function
mode
is the value in a data set that occurs most often. If all values occur the same amount of times there is no mode for that set.
domain
the set of all the input (x-values) for a function
scientific notation
a method of writing very large or very small numbers by using powers of 10
diameter
the distance across a circle through its center (is also twice the radius)
annual interest rate
apr, the percent of the principal you pay or earn a year
principal
the original amount of money loaned or borrowed
center
the point in the exact middle of a circle
irrational number
a number that can not be written a/b
volume
the amount of 3-dimensional space occupied by an object
least common multiple
the smallest multiple that two or more numbers have in common
theoretical probability
what should occur in a probability experiment...an experiment is not actually done
radical expression
an expression that contains a square root
Pythagorean triple
a set of three positive integers that work in the pythagorean theorem
equivalent ratios
Ratios that have the same value.
reciprocal
one of a pair of numbers whose product is 1: the reciprocal of 2/3 is 3/2
linear equation
an equation whose answers are ordered pairs (x,y) and whose answers graph a straight line
median
is the middle value in a data set where the numbers are ordered least to greatest
circumference
The distance around a circle.
proportion
an equation that states that two ratios are equal Ex: 1/2 = x/10
least common denominator
The least common multiple of the denominators of two or more fractions.
slope intercept form
y = mx + b
ratio
a comparison of two numbers by division
repeating decimal
a decimal in which one or more digits repeat infinitely
y-intercept
the point where a graph crosses the y-axis
Pi
3.1415.... is an irrational number resulting from the ratio of a circle's circumference to its diameter
slope
the steepness of a line on a graph, rise over run
function
a relation that assigns exactly one output value for each input value
experimental probability
probability based on what happens when an experiment is actually done
surface area
The total area of the 2-dimensional surfaces that make up a 3-dimensional object.
interest
is the amount paid for borrowing or lending money
rate
a ratio that compares two quantities measured in different units
perfect square
a number that has a whole number for a square root ex: 9, 16, 121
rational number
A number that can be written as a/b where a and b are integers, but b is not equal to 0.
multiple
skip-counting by any given number Ex: 4, 8, 12, 16...
probability of an event
number of favorable outcomes divided by total number of possible outcomes
circle
is the set of all points that are an equal distance from a point called the center
Pythagorean Theorem
a² + b² = c²
equivalent fractions
fractions that have the same value and the same simplest form
unit rate
a rate that has a denominator of 1
range
the set of all the output (y-values) for a function
Section 22
(53 cards)
In the equation y = 3x + 7, the slope is _______________
3
Lesson 1-1
Variables
Describe how you would graph y = 1/3 x + 2 using the slope and the y interept.
Put a dot on the y-axis on the number 2. Count a slope of 1/3 by going up 1 and right 3 or down 1 and left 3. Count the slope two or three times and then draw the line.
Look at the graph for problem #12 on page 218. What is the equation in slope intercept form?
y = −1/5 x + 1
Given the point (1, 9) and the slope 4, write the equation in slope intercept form
y = 4x + 5
Write the equation in slope intercept form:. . . y + 2 = 4(x + 2)
y = 4x + 6
What is true about the slopes of parallel lines?
The slopes of parallel lines are the same
dependent event (probability)
an event who's outcome does depend on the outcome of a previous event
combinations
...
What form should you put lines in to determine if they are parallel, perpendicular, or neither?
Slope intercept form
Determine whether the lines are parallel, perpendicular, or neither: . . . . . . . . . . . . . . . . . y = −2x and 2x + y = 3
Parallel
Write the equations for two lines that are parallel.
There are many answers. An example would by y = 2x and y = 2x + 7.
In the equation y = 3x + 7, the y intercept is _______________
7
Look at the graph for problem #42 on page 235. Write the equation in point slope form
y − 7 = −4/3(x + 3)
values of a variable
The numbers that can be represented by the variable.
Look at the graph for problem #40 on page 235. Write the equation in point slope form
y − 3 = 4(x − 1)
Given the point (2, 2) and m = −3, write the equation in point slope form
y − 2 = −3(x − 2)
Given the point (1, 3) and (−3, −5), write the equation in slope intercept form
y = 2x + 1
The point slope form of a linear equation is ________
y - y1 = m(x − x1)
Given the point (−8, 5) and m = −2/5, write the equation in point slope form
y − 5 = −2/5(x + 8)
In the equation y = −2x − 6, the slope is ________________
−2
Write the equation in standard form: y − 11 = 3(x − 2)
3x − y = −5
Title
Algebra Structure and Method Book 1
odds
the ratio of the number of ways the event can occur to the number of ways the event cannot occur. Favorable over Unfavorable.
Write an equation in slope intercept form for the line that passes through (3, 2) and is parallel r to y = x + 5.
y = x − 1
Chapter 1
Introduction to Algebra
What is true about any horizontal line and a vertical line?
They are perpendicular
In the equation, y = mx + b, the b stands for _____________
y-intercept
Write an equation in slope intercept form for the line that passes through (−1, −2) and is parallel to 3x − y = 5.
y = 3x + 1
Write an equation in slope intercept form for the line that passes through (−2, 2) and is perpendicular to y = −1/3 x + 9.
y = 3x + 8
tree diagram (probability)
a diagram used to show the total number of possible outcomes in an experiment
Write the equation in slope intercept form: 2x + 4y = 12
y = −1/2x + 3
Chapter 1 Section 1
Variables and Equations
Determine whether the lines are parallel, perpendicular, or neither: . . . . . . . . . . . . . . . . . 3x + 5y = 10 and 5x − 3y = −6
Perpendicular
In the equation, y = −2x − 6, the y intercept is _____________
−6
Write an equation in slope intercept form for the line that passes through (10, 5) and is perpendicular to 5x + 4y = 8.
y = 4/5x − 3
Write the equations for two lines that are perpendicular.
There are many answers. An example would by y = 2x and y = −1/2 x + 6. The product of the two slopes has to be −1
fundamental counting principle
...
What is true about the slopes of perpendicular lines?
The product of slopes of perpendicular lines is −1. (Slopes of perpendicular lines are opposite reciprocals)
Look at the graph for problem #34 on page 219. What is the equation in slope intercept form?
y = −4/7 x − 2
Write the equation in standard form: y − 10 = −(x − 2)
x + y = 12
The slope intercept form of a linear equation is ____
y = mx + b
In the equation, y = mx + b, the m stands for _____________
Slope
variable expression
An expression that contains a variable.
Look at the graph for problem #14 on page 218. What is the equation in slope intercept form?
y = −2 x + 3
variable
A symbol used to represent one or more numbers.
Determine whether the lines are parallel, perpendicular, or neither: . . . . . . . . . . . . . . . . . 2x + 5y = 15 and 3x + 5y = 15
Neither
independent event (probability)
an event that is not affected by another event.
Objective 1-1
To simplify numerical expressions and evaluate algebraic expressions.
permutations
...
Section 23
(53 cards)
D = rt
Distance traveled = rate × time traveled
substitution principle
An expression may be replaced by another expression that has the same value.
Lesson 1-4
Translating Words into Symbols
solve an open sentence
To find the solution set of the sentence.
origin
The zero point on a number line. The intersection of the axes on a coordinate plane.
sides of an equation
The two expressions joined by the equals sign.
numerical expression
An expression that names a particular number; a numeral.
Objective 1-4
To translate phrases into variable expressions.
Lesson 1-5
Translating Sentences into Equations
value of a numerical expression
The number named by the expression.
perimeter
The perimeter of a plane figure is the distance around it.
solution set of an open sentence
The set of all solutions of the sentence.
Plan for Solving a Word Problem Step 4
Solve the equation and find the unknowns asked for.
Plan for Solving a Word Problem Step 3
Reread the problem and write an equation that represents relationships among the numbers in the problem.
Objective 1-3
To find solution sets of equations over a given domain.
formula
An equation that states a rule about a relationship.
Chapter 1 Section 3
Numbers on a Line
solution of a sentence
Any value of a variable that turns an open sentence into a true statement.
When there are no grouping symbols, simplify in the following order:
1. Do all multiplications and divisions in order from left to right. 2. Do all additions and subtractions in order from left to right.
Plan for Solving a Word Problem Step 1
Read the problem carefully. Decide what unknown numbers are asked for and what facts are known. Making a sketch may help.
Objective 1-7
To use the five-step plan to solve word problems over a given domain.
Plan for Solving a Word Problem Step 5
Check your results with the words of the problem. Give the answer.
Plan for Solving a Word Problem Step 2
Choose a variable and use it with the given facts to represent the unknowns described in the problem.
Lesson 1-2
Grouping Symbols
numeral
An expression that names a particular number; a numerical expression.
A = lw
Area of rectangle = length of rectangle × width of rectangle
Chapter 1 Section 2
Applications and Problem Solving
Step 2
Choose a variable and represent the unknows.
Objective 1-2
To simplify expressions with and without grouping symbols.
negative side
On a horizontal number line, the side to the left of the origin.
evaluating a variable expression
Replacing each variable in the expression by a given value and simplifying the result.
P = 2l + 2w
Perimeter of rectangle = ( 2 × length ) + ( 2 × width )
Step 3
Reread the problem and write an equation.
Lesson 1-6
Translating Problems into Equations
positive side
On a horizontal number line, the side to the right of the origin.
satisfy an open sentence
Any solution of the sentence satisfies the sentence.
Lesson 1-8
Number Lines
Lesson 1-7
A Problem Solving Plan
C = np
Cost = number of items × price per item
simplifying a numerical expression
Replacing the expression by the simplest name for its value.
equation
A statement formed by placing an equals sign between two numerical or variable expressions.
Objective 1-5
To translate word sentences into equations.
area
The area of a region is the number of square units it contains.
open sentence
A sentence containing one or more variables.
Objective 1-8
To graph real numbers on a number line and to compare real numbers.
domain of a variable
The given set of numbers that the variable may represent.
Step 1
Read the problem carefully.
grouping symbol
A device used to enclose an expression that should be simplified before other operations are performed. Examples: parentheses, brackets, fraction bar.
Objective 1-6
To translate simple word problems into equations.
Lesson 1-3
Equations
Section 24
(52 cards)
positive number
A number paired with a point on the positive side of a number line.
positive integers
The numbers 1, 2, 3, 4, and so on.
height
the vertical dimension of extension
dimensions
The length, width, and height of an object being measured.
formula
mathematics a standard procedure for solving a class of mathematical problems
integers
The set consisting of the positive integers, the negative integers, and zero.
intercept
the point at which a line intersects a coordinate axis
dependent
the variable in the relation whos value depends on the value of the independent variable
Zero is niether
positive nor negative.
centimeter
a metric unit of length equal to one hundredth of a meter
dilation
A transformation that changes the size of an object, but not the shape.
3. If a = 0,
then −a = 0.
congruent
Having the same size and shape
base
in an expression of the form x m the base is x
1. If a is positive,
then −a is negative.
bisect
to cut or divide into two equal parts: to bisect an angle.
bisector
a point, ray, line, line segment, or plane that intersects the segment at its midpoint
real number
Any number that is either positive, negative, or zero.
Objective 1-9
To use opposites and absolute values.
graph
a drawing illustrating the relations between certain quantities plotted with reference to a set of axes
decrease
the amount by which something decreases
cubes
a number that is a whole number raised to the third power Ex. 8, 27, 64, 125, etc.
negative integers
The numbers −1, −2, −3, −4, and so on.
graph of a number
The point on a number line that is paired with the number.
domain
the set of values of the independent variable for which a function is defined
end point
the point in a titration at which a marked color change takes place
2. If a is negative,
|a| = −a.
grid
a network of horizontal and vertical lines that provide coordinates for locating points on an image
absolute value
The positive number of any pair of opposite nonzero real numbers is the absolute value of each number in the pair. The absolute value of 0 is 0. The absolute value of a number a is denoted by |a|.
axis
a line about which a three-dimensional body or figure is symmetrical.
2. If a is negative,
then −a is positive.
inequality symbols
Symbols used to show the order of two real numbers. The symbol ≠ means "is not equal to."
3. If a is zero,
|a| = 0.
constant
a number representing a quantity assumed to have a fixed value in a specified mathematical context
Lesson 1-9
Opposites and Absolute Values
opposite of a number
Each of the numbers in a pair such as 6 and −6 or −2.5 and 2.5. Also called additive inverse.
expression
a group of symbols that make a mathematical statement
diameter
the length of a straight line passing through the center of a circle and connecting two points on the circumference
coordinate
a number that identifies a position relative to an axis
negative number
A number paired with a point on the negative side of a number line.
1. If a is positive,
|a| = a.
coordinate of a point
The number paired with that point on a number line.
function
a mathematical relation such that each element of one set is associated with at least one element of another set
whole numbers
The set consisting of zero and all the positive integers.
cylinders
a three dimensional figure with two parallell, congruent circular basis connected by a curved lateral surface
hexagon
a six-sided polygon
4. The opposite of −a is
a; that is, −(−a) = a.
estimate
an approximate calculation of quantity or degree or worth
increase
the amount by which something increases
diagram
A visual representation of data to help readers better understand relationships among data
Section 25
(54 cards)
linear
designating or involving an equation whose terms are of the first degree
tables
Used to arrange text in columns and rows
three dimensional
Having the dimensions of height, width, and depth.
minimum
the point on a curve where the tangent changes from negative on the left to positive on the right
reduced
made less in size or amount or degree
reflections
flips a figure over a line
parallelogram
a quadrilateral whose opposite sides are both parallel and equal in length
length
the linear extent in space from one end to the other
rotation
a single complete turn axial or orbital
tax rate
the amount of tax people are required to pay per unit of whatever is being taxed
quadrilaterals
4 sided polygon
semi circle
half of a circle
intersection
the act of intersecting as joining by causing your path to intersect your target's path
symmetry
an attribute of a shape or relation
milligrams
used to measure the mass of very small objects one thousandths
interval
a set containing all points or all real numbers between two given endpoints
matrix
a rectangular array of elements or entries set out by rows and columns
value
a numerical quantity measured or assigned or computed
segments
set of points on a line that consist of two points called endpoints, and all the points between them
pyramid
a polyhedron having a polygonal base and triangular sides with a common vertex
translation
the act of changing in form or shape or appearance
range
the difference between the highest and lowest scores in a distribution
rectangle
a parallelogram with four right angles
plane
an unbounded two-dimensional shape
pendulum
a weight, hanging from a point, that swings an equal distance from side to side; often used in clocks
solutions
ways to solve problems
ordered pair
A pair of numbers, x, y, that indicate the position of a point on a Cartesian plane.
inverse
opposite in nature or effect or relation to another quantity
quadratic equations
a function that has the variable raised to the second power
secant
ratio of the hypotenuse to the adjacent side of a right-angled triangle
meter
any of various measuring instruments for measuring a quantity
polygon
closed plane figure having, literally, many angles and therefore many sides
modeled
resembling sculpture
slope
the steepness of a line on a graph, equal to its vertical change divided by its horizontal change
non-collinear
points that do not lie on the same line
quantity
something that has a magnitude and can be represented in mathematical expressions by a constant or a variable
Absolute Value
The distance a number is from the 0 on the number line
Algebraic Expression
An expression that contains numbers, operations and variables.
narrowest
the field of vision is a cone-shaped area with its narrowest/widest end near the driver
squares
all sides are equal, opposite sides are parallel, all angles are 90 degrees
relations
a set of ordered pairs
perimeter
the size of something as given by the distance around it
isosceles triangle
a triangle with at least two congruent sides
volume
the amount of 3-dimensional space occupied by an object
scale factor
The ratio of the lengths of two corresponding sides of two similar polygons
level
having a horizontal surface in which no part is higher or lower than another
maximum
the point on a curve where the tangent changes from positive on the left to negative on the right
point
a geometric element that has position but no extension
widest
The widest/narrowest part of the nuchal translucency should be measured.
perpendicular
intersecting at or forming right angles
Section 26
(55 cards)
Coordinate Plane
A coordinate system formed by the intersection of a horizontal number line, called the x-axis, and a vertical number line, called the y-axis.
What do you do when you have a negative denominator? ex b-2/-5
Move it up top. answer: 2-b/5 (Make sure to change signs)
Dividend
a number to be divided by another number
Unit Conversion
The process of renaming a measurement using different units.
Grouping Symbols
parentheses ( ), brackets [ ], and braces { } that group parts of an expression.
Constant
A term that has no variable.
Greatest Common Factor
The greatest factor that is common to two or more numbers.
Equivalent fractions
fractions that have the same value
How would you describe this inequality in interval notation: x ≥ 2 ?
[ 2 , ∞ )
What is the first step in this equation : 2|2x+1| -4 = 16 ?
Get rid of the variables outside of the absolute value.
Is this a linear equation or a linear function: f(x) = 2x - 1 ?
A linear function
Factors
Whole numbers that can be multiplied together to hind a product.
What is the solution for |x - 5| ≥ 0?
R. Any real number, because the absolute value of a number is always ≥ 0.
Base of a Power
The repeated factor in a power.
For what inequality symbols do you use brackets?
≤ and ≥
How do you find the x-intercept in a linear equation? ax + by = c
Set y = 0 to isolate the x.
Evaluate
to find the value of a numerical or algebraic EXPRESSION.
How would you describe this inequality in a solution set: x < 5 ?
{x|x<5}
Improper Fraction
A fraction whose numerator is greater than or equal to its denominator.
What is the standard form for a linear equation?
ax + by = c Both x and y are on the right side of the equation.
Irrational numbers
numbers that cannot be expressed in the form a/b, where a and b are integers and b =0.
What is the solution for |x - 5| ≤ 0?
{5} Absolute value of a number can never be < 0 but it can be 0 when x =5. |5 - 5| = 0
For what inequality symbols do you use parenthesis?
< and >
Equivalent expressions
algebraic expressions that have the same values for all values of variables
Equation
a mathematical statement in which two expressions are equal
Describe this inequality in interval notation: x < 5/3 ?
( -∞ , 5/3)
Distributive Property
a(b+c)=ab+ac, Multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products.
How would you word the function (front) part of this equation? P(x) = 7.25x
P is a function of x
What is the solution for |x - 5| < 0?
∅ The absolute value of a number can never be < 0. No solution.
What will the solution of this absolute value equation look like |exp.| > a ?
exp. < -a or exp. > a
Decimal
a number with one or more digits to the right of the decimal point
Is this a linear equation or a linear function: y = 2x - 1 ?
A linear equation.
How do you find the y-intercept in a linear equation? ax + by = c
Set x = 0 to isolate the y.
Divisor
the number by which a dividend is divided.
What do you do when you divide a negative number in an inequality?
Switch the signs
Exponent
The number of times a factor in repeated in a power.
Formula
An algebraic equation that shows the relationship among specific quantities.
Elimination Method
to solve a system of equations by adding equations to get rid of one variable. sometimes needing to multiply one equation by a # to make terms opposites
Function
A pairing of input and output values according to a specific rule.
Associative Property
A property that states that numbers in addition of multiplication expressions can be grouped without affecting the value of the expression.
What will the solution of this absolute value equation look like |exp.| = a ?
exp = a or exp = -a
Coefficient
The number mulltiplied by a variable in a term.
Inverse Operations
Operations that undo each other.
Integers
The set of all whole numbers, their opposites, and 0.
The graph of y = b is?
A horizontal line through b on the y axis. (Slope = 0)
How would you express this inequality in a solution set: -4 < t ≤ 5/3 ?
{t| -4 < t ≤ 5/3}
Axes
A horixontal and vertical number line on a coordinate plane.
Least Common Denominator
The least common multiple of the denominators of two or more fractions.
Commutative Property
A property that states numbers can be added or multiplied in any order.
What will the solution of this absolute value equation look like |exp.| < a ?
-a < exp < a
Section 27
(54 cards)
When graphing linear inequalities, for what symbols would you use a solid line in the graph?
≤ and ≥
variable
A symbol used to represent one or more numbers.
Lesson 1-3
Equations
Objective 1-3
To find solution sets of equations over a given domain.
How does slope tell us the tilt? y/x
Because y/x = rise/run
How do we view lines on the graph to get the correct slope?
Left to right
How do you find the equation of a line if you are given two points, but no y-intercept or slope? Such as: (4 , 3) (6 , -2)
You would have to find the slope first, x² - x¹/y² - y¹, and then use the point slope equation, y - y¹ = m(x - x¹)
A linear system where two lines are directly on top of each other (infinite answers) would be what type of system?
A dependent system.
Chapter 1 Section 1
Variables and Equations
How do we calculate slope from two given points? (x¹,y¹) (x²,y²)
y² - y¹/x² - x¹ to get the slope. (rise/run)
When graphing linear inequalities, for what symbols would you use a dotted line in the graph?
< and >
sides of an equation
The two expressions joined by the equals sign.
These two equations would be easiest to solve by what method Equation¹: 4x + 2y = 6 Equation²: 3x - 2y = 16?
The addition method. If set up as if adding them together the y's would cancel. Then you could solve for x and then solve for y.
simplifying a numerical expression
Replacing the expression by the simplest name for its value.
A linear system where two lines are parallel (no answer) would be what type of system?
An inconsistent system.
value of a numerical expression
The number named by the expression.
open sentence
A sentence containing one or more variables.
What is the formula for slope-intercept form?
y = mx + b
What is the point slope equation? Used when you have a slope and a random point.
y - y¹ = m(x - x¹)
Objective 1-1
To simplyfy numerical expressions and evaluate algebraic expressions.
When there are no grouping symbols, simplify in the following order:
1. Do all multiplications and divisions in order from left to right. 2. Do all additions and subtractions in order from left to right.
numeral
An expression that names a particular number; a numerical expression.
Chapter 1
Introduction to Algebra
variable expression
An expression that contains a variable.
How do you graph the line of a linear inequality? (Also the 2nd step) y ≥ 1/2x - 3
Set the inequality as an equation y ≥ 1/2x - 3 turns into y = 1/2x - 3, and plot the line.
How should you decide what side will be shaded in while graphing a linear inequality? y ≥ 1/2x - 3
Test it by putting in a point, typically (0 , 0) and see if it's true. 0 ≥ 1/2(0) -3 is 0 ≥ 3, which means true. Shade on the side of the point. If false shade on opposite side.
grouping symbol
A device used to enclose an expression that should be simplified before other operations are performed. Examples: parentheses, brackets, fraction bar.
numerical expression
An expression that names a particular number; a numeral.
These two equations would be easiest to solve by what method Equation¹: y = -2x + 6 Equation²: 3x - 2y = 16?
The substitution method, substitue first equation in place of y in the second: 3x - 2(-2x + 6) = 16 and solve.
What is the equation of the line through (0 , 2) that is perpendicular to the graph of the line: y = 1/2x - 4 ? (Think of the perpendicular rule)
y = -2/1x + 2 because (0 , 2) gives us the y intercept and -2/1 is the reciprocal of the line in the question.
evaluating a variable expression
Replacing each variable in the expression by a given value and simplifying the result.
equation
A statement formed by placing an equals sign between two numerical or variable expressions.
How is a positive line viewed?
It runs upwards. m>0
Lesson 1-1
Variables
What three methods could be used to solve systems of linear equations? Equation¹: 2x + y = 6 Equation²: 3x - 2y = 16
By graphing, the substitution method, and the addition method.
When solving a linear system of equations, if the substitution or addition method resulted in 2 ≠ 4, what system would you have?
An inconsistent system, parallel lines. No solution.
substitution principle
An expression may be replaced by another expression that has the same value.
Lesson 1-2
Grouping Symbols
A linear system where two lines cross (one answer) would be what type of system?
A consistent system.
Objective 1-2
To simplify expressions with and without grouping symbols.
When solving a linear system of equations, if the substitution or addition method resulted in 2 = 2, what would you have?
A dependent system, two lines on top of each other. ∞ Solutions
What is the first step in graphing a linear inequality? y ≥ 1/2x - 3
Decide what type of line it will be. In this case it's a solid line because of the ≥ sign.
How is slope defined?
The slope of a line measures the tilt of the line.
What is the difference in slopes for perpendicular lines?
They have negative reciprocals. ex. Line¹: m= 2/4, would mean Line²: m= -4/2
If m=0 what type of line would be viewed on the graph?
A horizontal line.
How is a negative line viewed?
It runs downwards. m<0
What do parallel lines have in common?
Equal slopes. m¹ = m²
Title
Algebra Structure and Method Book 1
The graph of x = b is?
A vertical line through b on the x axis. (Slope undefined)
values of a variable
The numbers that can be represented by the variable.
Section 28
(54 cards)
solution of a sentence
Any value of a variable that turns an open sentence into a true statement.
Lesson 1-7
A Problem Solving Plan
negative number
A number paired with a point on the negative side of a number line.
Lesson 1-5
Translating Sentences into Equations
Lesson 1-9
Opposites and Absolute Values
Zero is niether
positive nor negative.
Plan for Solving a Word Problem Step 5
Check your results with the words of the problem. Give the answer.
perimeter
The perimeter of a plane figure is the distance around it.
Plan for Solving a Word Problem Step 2
Choose a variable and use it with the given facts to represent the unknowns described in the problem.
Step 1
Read the problem carefully.
A = lw
Area of rectangle = length of rectangle × width of rectangle
1. If a is positive,
then −a is negative.
Objective 1-5
To translate word sentences into equations.
Step 2
Choose a variable and represent the unknows.
integers
The set consisting of the positive integers, the negative integers, and zero.
solve an open sentence
To find the solution set of the sentence.
Objective 1-4
To translate phrases into variable expressions.
positive integers
The numbers 1, 2, 3, 4, and so on.
Lesson 1-6
Translating Problems into Equations
D = rt
Distance traveled = rate × time traveled
negative integers
The numbers −1, −2, −3, −4, and so on.
Objective 1-8
To graph real numbers on a number line and to compare real numbers.
Plan for Solving a Word Problem Step 3
Reread the problem and write an equation that represents relationships among the numbers in the problem.
origin
The zero point on a number line. The intersection of the axes on a coordinate plane.
negative side
On a horizontal number line, the side to the left of the origin.
satisfy an open sentence
Any solution of the sentence satisfies the sentence.
P = 2l + 2w
Perimeter of rectangle = ( 2 × length ) + ( 2 × width )
Plan for Solving a Word Problem Step 1
Read the problem carefully. Decide what unknown numbers are asked for and what facts are known. Making a sketch may help.
Chapter 1 Section 2
Applications and Problem Solving
whole numbers
The set consisting of zero and all the positive integers.
area
The area of a region is the number of square units it contains.
C = np
Cost = number of items × price per item
positive number
A number paired with a point on the positive side of a number line.
real number
Any number that is either positive, negative, or zero.
Step 3
Reread the problem and write an equation.
graph of a number
The point on a number line that is paired with the number.
Chapter 1 Section 3
Numbers on a Line
coordinate of a point
The number paired with that point on a number line.
inequality symbols
Symbols used to show the order of two real numbers. The symbol ≠ means "is not equal to."
Objective 1-7
To use the five-step plan to solve word problems over a given domain.
Objective 1-6
To translate simple word problems into equations.
solution set of an open sentence
The set of all solutions of the sentence.
positive side
On a horizontal number line, the side to the right of the origin.
Plan for Solving a Word Problem Step 4
Solve the equation and find the unknowns asked for.
opposite of a number
Each of the numbers in a pair such as 6 and −6 or −2.5 and 2.5. Also called additive inverse.
Lesson 1-4
Translating Words into Symbols
domain of a variable
The given set of numbers that the variable may represent.
formula
An equation that states a rule about a relationship.
Objective 1-9
To use opposites and absolute values.
Lesson 1-8
Number Lines
Section 29
(52 cards)
y = ±√(−x+3)
y = ±√(x+3)
c) (x⁴−9)/(x²+1)
Which function is shown in blue? a) (x⁴−9)/(x²+9) b) (x⁴−9)/(x²+2) c) (x⁴−9)/(x²+1)
y = ±√x
y = −x²
y = x² + 3
y = −5x²
y = − (1/5)x²
When dividing two terms with the same base, you should _____ the exponents.
subtract
y = (x+2)²
y = x² + 3x
What is the degree of the polynomial h(t) = -8t² + 5 - 3t³?
The degree is 3
1. If a is positive,
|a| = a.
y = (x − 3)²
y = x² − 9
3. If a is zero,
|a| = 0.
When multiplying two terms with the same base, you should ____ the exponents.
add
y = (x−2)²
y = −x² −2x
What is the greatest common mononial factor of 9x³y² + 15x²y - 6xy² ?
3xy
What is the end behavior for f(x) = 3x⁴ - x² +1 ?
Both sides up.
3. If a = 0,
then −a = 0.
y = ±√x + 3
y = (x−1)² +3
y = ±√x − 5
absolute value
The positive number of any pair of opposite nonzero real numbers is the absolute value of each number in the pair. The absolute value of 0 is 0. The absolute value of a number a is denoted by |a|.
y = x² + 10
What is the end behavior for f(x) = -2x³ + x² - 2?
Left side up, right side down.
y = (x+3)² −5
b = 4
This graph does NOT represent ax² -1/4, where a would be a fraction less than 1. This graph is of the form (x⁴ - a)/(x² + b) Find the integer, b. HINT Type "b= ..."
y = (x−2)² +5
y = x² − 4
If x-2 is a factor of a polynomial f(x), is it true that f(2) = 0 ?
True.
4. The opposite of −a is
a; that is, −(−a) = a.
y = x² − 3
y = ±√−x + 2
y = x² −3x
y = (x+1)²
2. If a is negative,
|a| = −a.
y = ±√−x
y = ± √(x − 5)
2. If a is negative,
then −a is positive.
y = x² −25
y = x²
What is (3.2 x 10^5)(1.4 x 10^-2) written in scientific notation?
4.48 x 10³
If x + 3 is a factor of x³ − x² − 17x − 15, what are the other factors?
(x + 1) and (x - 5)
Is ± ½ a possible rational solution of f(x)= - 3x³ - 11x² + 5x - 6?
No.
y = x² −5x
y = (x+5)²
What is the complete factorization of 3x⁴ - 3x² ?
3x²(x-1)(x+1)
Section 30
(55 cards)
A zero of a function is also the _____ on a graph?
x-intercept
Elimination Method
to solve a system of equations by adding equations to get rid of one variable. sometimes needing to multiply one equation by a # to make terms opposites
Factor the polynomial completely: 3x³ + 6x² + x + 2
(3x²+1)(x+2)
Associative Property
A property that states that numbers in addition of multiplication expressions can be grouped without affecting the value of the expression.
Divisor
the number by which a dividend is divided.
Determine the possible number of positive real zeros, negative real zeros, and imaginary zeros for the function: f(x) = x⁴ + 3x³ - 2x² - x + 10
positive zeros: 0 or 2, negative zeros: 0 or 2, imaginary zeros 0, 2, or 4
Absolute Value
The distance a number is from the 0 on the number line
Exponent
The number of times a factor in repeated in a power.
Factors
Whole numbers that can be multiplied together to hind a product.
Decimal
a number with one or more digits to the right of the decimal point
Coefficient
The number mulltiplied by a variable in a term.
Improper Fraction
A fraction whose numerator is greater than or equal to its denominator.
Algebraic Expression
An expression that contains numbers, operations and variables.
Determine the possible number of positive real zeros, negative real zeros, and imaginary zeros for the function: g(x) = -x^5 + 2x⁴ + 3x² - 7x - 12
positive zeros: 0 or 2, negative zeros: 1, imaginary zeros: 2 or 4
Using long division: (x⁴ + 10x³ + 8x² - 59x +40) ÷ (x² + 3x -5)
x² + 7x - 8
Function
A pairing of input and output values according to a specific rule.
Find all real zeros of the function: f(x) = x³ - 3x² - x +3
-1, 1, 3
Commutative Property
A property that states numbers can be added or multiplied in any order.
Simplify (¾)^-3
64/27
Grouping Symbols
parentheses ( ), brackets [ ], and braces { } that group parts of an expression.
Equivalent expressions
algebraic expressions that have the same values for all values of variables
Evaluate
to find the value of a numerical or algebraic EXPRESSION.
At which value of x does f(x) = 2x³ - x² + 1 have a local minimum value?
(0,1)
Which of the following, based on the Descartes Rule of Signs, is the only possible classification of the roots of the function f(k) = -3k³ + 5k² - k + 4?
3, 1 positives; 0 negatives; 0, 2 imaginary
Base of a Power
The repeated factor in a power.
Unit Conversion
The process of renaming a measurement using different units.
Formula
An algebraic equation that shows the relationship among specific quantities.
Coordinate Plane
A coordinate system formed by the intersection of a horizontal number line, called the x-axis, and a vertical number line, called the y-axis.
(3³)² is the power of power property. It tells us to do what to the exponents?
Multiply exponents, so 3^5.
Axes
A horixontal and vertical number line on a coordinate plane.
Perform the indicated operation: (2x⁴+9x-7) - (x⁴+6x+5)
x⁴ + 3x - 12
Using synthetic division: (2x³ - 25x² + 83x - 88) ÷ (x-8)
2x² - 9x + 11
Find all zeros of the polynomial function: h(x) = 2x⁴ - 3x³ - 27x² + 62x - 24
-4, ½, 2, 3
Integers
The set of all whole numbers, their opposites, and 0.
When multiplying binomials, what method do we use?
FOIL
Find all real zeros of the function: f(x) = x³ -6x² +4x - 24
6
Inverse Operations
Operations that undo each other.
Factor the polynomial completely: 3x³-81
3(x-3)(x²+3x+9)
Find all zeros of the polynomial function: g(x) = x³ - 2x² - x + 2
-1, 1, 2
Equation
a mathematical statement in which two expressions are equal
Perform the indicated operation: (7x - 3)²
49x² - 42x + 9
Distributive Property
a(b+c)=ab+ac, Multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products.
Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros: -3, ± √2
f(x) = x³ + 3x² - 2x -6
Equivalent fractions
fractions that have the same value
Find all real zeros of the function: f(x) = x⁴ - 2x³ - 8x² + 8x + 16
-2, 2, 1 ± √5
Perform the indicated operation: (x - 2)(x+3)(x-5)
x³ - 4x² - 11x + 30
Greatest Common Factor
The greatest factor that is common to two or more numbers.
Constant
A term that has no variable.
Dividend
a number to be divided by another number
What does FOIL stand for?
First, Outter, Inner, Last
Section 31
(52 cards)
pivot column
A column that contains a pivot position
Properties of transposition
1. (A^T)^T = A; 2. (A+B)^T = A^T + B^T; 3. (rA)^T = rA^T; 4. (AB)^T = B^TA^T
Intersect
when two lines cross
Echelon form
1. All nonzero rows are above any all zero rows; 2. Each leading entry is in a column to the right of the previous leading entry; 3. All entries below a leading entry in its column are zeros
Triangle
a polygon with 3 sides and vertices
leading entry
Leftmost non-zero entry in a non-zero row
Dimension
The number of vectors in any basis of H; the zero subspace's dimension is 0
Least Common Multiple
...
Invertibility rules
1. If A is invertible, (A^-1)^-1 = A; 2. (AB)^-1 = B^-1 * A^-1; 3. (A^T)^-1 = (A^-1)^T
Column space
Set of all the linear combinations of the columns of A
Straight Angle
two right angles that forms a straight line back to back
Acute Angle
an angle smaller than a right angle
Leontief input-output model
x = Cx + d
inner product
a matrix product u^Tv or u . v where u and v are vectors; if U . V = 0, u and v are orthogonal
orthonormal
An orthogonal set of unit vectors
Matrix multiplication warnings
1. AB != BA ; 2. If AB = AC, B does not necessarily equal C; 3. If AB = 0, it cannot be concluded that either A or B is equal to 0
Right Angles
the angles made by perpendicular lines
Ax = b
1. For each b in R^n, Ax = b has a solution; 2. Each b is a linear combination of A; 3. The columns of A span R^n; 4. A has a pivot position in each row
linear equation
An equation that can be written as a1x1 + a2x2 + ... = b; a1, a2, etc. are real or complex numbers known in advance
Column Row Expansion of AB
col1Arow1B + ...
one-to-one
A transformation that assigns a vector y in R^m for each x in R^n; there's a pivot in every column
Basis
A linearly independent set in H that spans H; the pivot columns of A form a basis for A's column space
Span
the collection of all vectors in R^n that can be written as c1v1 + c2v2 + ... (where c1, c2, etc. are constants)
Reduced Echelon Form
Same as echelon form, except all leading entries are 1; each leading 1 is the only non-zero entry in its row; there is only one unique reduced echelon form for every matrix
Transposition
flips rows and columns
LU Factorization
1. Ly = b; Ux = y; 2. Reduce A to echelon form; 3. Place values in L that, by the same steps, would reduce it to I
inconsistent system
Has no solution
Perpendicular
when two lines make square corners at the point of intersection
Least Common Denominator
The least common multiple of the denominators of two or more fractions.
Polygon
simple, closed, flat geometric figures whose sides are line segments and whose lines do not cross
rank
The dimension of the column space
Obtuse Angle
an angle larger than a right angle
Line Segment
part of a line
Parallel Lines
lines in the same plane that do not intersect and the distance between the lines is always the same
Line
straight line that has no width and no ends
consistent system
Has one or infinitely many solutions
Irrational numbers
numbers that cannot be expressed in the form a/b, where a and b are integers and b =0.
Subspaces
1. The zero vector is in H; 2. For u and v in H, u + v is also in H; 3. For u in H, cu is also in H (c is a constant)
orthogonal set
A set of vectors where Ui . Uj = 0 (and i != j); if S is an orthogonal set, S is linearly independent and a basis of the subspace spanned by S
transformation
assigns each vector x in R^n a vector T(x) in R^m
independent
If only the trivial solution exists for a linear equation; the columns of A are independent if only the trivial solution exists
Point of Intersection
the place where two lines cross
Mixed Number
the sum of a whole number and a fraction
homogeneous
A system that can be written as Ax = 0; the x = 0 solution is a TRIVIAL solution
Invertible Matrix Theorem (either all of them are true or all are false)
A is invertible; A is row equivalent to I; A has n pivot columns; Ax = 0 has only the trivial solution; The columns of A for a linearly independent set; The transformation x --> Ax is one to one; Ax = b has at least one solution for each b in R^n; The columns of A span R^n; x --> Ax maps R^n onto each R^m; there is an n x n matrix C such that CA = I; there is a matrix such that AD = I; A^T is invertible; The columns of A form a basis of R^n; Col A = R^n; dim Col A = n; rank A = n; Nul A = [0]; dim Nul A = 0
onto
consistent for any b; pivots in all rows
pivot position
A position in the original matrix that corresponds to a leading 1 in a reduced echelon matrix
dependent
If non-zero weights that satisfy the equation exist; if there are more vectors than there are entries
orthogonal component
1. x is in W' if x is perpendicular to every vector that spans W; 2. W' is a subspace of R^n
Null space
Set of all solution to Ax = 0
Section 32
(55 cards)
Trapezoid
a quadrilateral that has exactly two parallel sides
Regular Polygons
when all the sides of a polygon are the same length and all the angles are the same measure
Natural Numbers
counting numbers
Octagon
a polygon with 8 sides and vertices
Concave
an indentation
Concave Polygon
a polygon with an indentation
Dodecagon
a polygon with 12 sides and vertices
Positive Real Numbers
any number that can be used to describe a physical distance greater than zero
Heptagon
a polygon with 7 sides and vertices
Decagon
a polygon with 10 sides and vertices
Equianglular Triangle
a triangle in which all the angles are the same measure
Symbol of Inequality
the unequal sign which indicates that two quantities are not equal
Value
the number represented by a numeral
Circumference
the perimeter of a circle
Real Numbers
all the positive real numbers plus zero
Signed Numbers
numbers that are indicated with signs of postive or negative used in algebraic addition
Undecagon
a polygon with 11 sides and vertices
Equilateral Triangle
a triangle in which the length of all the sides are equal
Nonagon
a polygon with 9 sides and vertices
Convex Polygon
any polygon without an indentation
Factor
the numbers multiplied in a multiplication problem
Right Triangle
a triangle with a right angle
Number
an idea that represents quantity
Parallelogram
a quadrilateral that has two pairs of parallel sides
Radius
the distance from the center of the circle to the outside edge
Minuend
the first number in a subtraction problem
Numeral
a symbol which represents the idea of a particular number
Sum
the result of an addition problem
Rectangle
a parallelogram with four right angles
Perimeter
the measure around
Addend
the numbers added in an addition problem
Rhombus
an equilateral parallelogram
Equilateral Polygon
when all the sides of a polygon have the same length
Difference
the result of a subtraction problem
Square
a rhombus with four right angles
Subtrahend
the second number in a subtraction problem
Diameter
the distance from one side of a circle through the center to the other side
Acute Triangle
a triangle with angles smaller than 90 degrees
Irrational Number
a number that would take an infinite number of digits to express
Equiangular Polygon
when all the angles of a polygon have the same measure
Tick Marks
marks used to denote sides and angles that have the same length and/or measure
Symbol of Equality
the equal sign which indicates that two quantities are equal
Isoceles Triangle
a triangle that has at least two sides of equal length
Obtuse Triangle
a triangle with an angle that is greater than 90 degrees
Zero
a number that can be used to describe a physical distance of no magnitude
Negative Numbers
the negative counterparts of the positive real numbers
Pentagon
a polygon with 5 sides and vertices
Quadrilateral
a polygon with 4 sides and vertices
Hexagon
a polygon with 6 sides and vertices
Scalene Triangle
a triangle in which all sides are different
Section 33
(50 cards)
⁵√x⁵
x
Numerator
the number on the top of a fraction
radicand of ⁵√x
x
y varies inversely as x (k is the constant)
y=k/x
(g⁴)²
g⁸
intersection, includes only the numbers in common from 2 sets, x<5, x>2 →includes only numbers less than 5 but greater than 2 (2, 5)
∩
side
Each segment of a polygon.
union, includes all possibilities from both sets, x<5, x>2 →includes all real numbers(⁻∞, ∞)
∪
Dividend
the first number in a division problem
(p³)⁴
p¹²
2x⁻² /x³
2/x⁵
point of intersection
The point where two lines cross.
index of ⁵√x
5
Divisor
the second number in a division problem
x¹/³
³√x
greater than or equal to x, less than or equal to y
[x, y]
Product
the result of a multiplication problem
mixed number
The sum of a whole number and a fraction.
2x² /x³
2/x
Q is directly proportional to y and inversely proportional to y (k is the constant)
Q=kx/y
no x value in the domain can be paired with more than one y value, vertical line test,
function
factor a²-b²
(a+b)(a-b)
(a-b)(a-b) OR (a - b)²
a² -2ab + b²
h⁷/⁹ ÷ h⁵/⁹
h²/⁹
factor 8m²+64
8(m²+8)
acute angle
An angle that is smaller than a right angle.
domain in interval notation f(x)=⁵√x-5 (for an odd index the root can be negetive or positive)
(-∞,∞)
Quotient
the result of a division problem
(a+b)(a+b) OR (a+b)²
a²+2ab+b²
vertex form
f(x) = a(x-h)² + k, where (h,k) is the vertex form of quadratic equation
greater than x, less than y in interval notation
(x, y)
⁴√x⁴
|x|
Denominator
the number on the bottom of a fraction
x²/³
(³√x)²
domain in interval notation f(x)=⁴√x-5 ( for an even index, root must be positive)
[5,∞)
factor a³-b³
(a-b)(a²+ab+b²)
straight angle
Two right angles form a straight angle.
7c¹/⁶
7 ⁶√c
quadratic formula
x = -b ± √(b² - 4ac)/2a
vertex formula
(-b/2a, 4ac-b²/4a)
right angle
The angle made by perpendicular lines.
⁴√7 * ⁴√2
⁴√14
Determine the domain and write your answer in set-builder notation. v+3/v-5
{v|v≠5}
y varies directly as x (k is the constant)
y=kx
x⁻⁵
1 / x⁵
factor a³+b³
(a+b)(a²-ab+b²)
perpendicular
Two lines that make a square corner at the point of intersection.
x⁻⁵/⁶
1 / ⁶√x⁵
vertex
The endpoint of each segment in a polygon.
obtuse angle
An angle that is larger than a right angle but less than a straight angle.
Section 34
(50 cards)
equiangular polygon
A polygon in which all angles have the same measure.
real number
All negative or positive numbers and zero.
sum
The result of an addition problem.
right triangle
A triangle that has a right angle.
decimal system
The system of numeration that is used to designate numbers.
convex polygon
Any polygon that does not have an indentation. Most polygons you will study are convex polygons
180 degrees
The sum of measures of the three angles in any triangle is 180 degrees.
negative real number
Any number that can be used to describe the negative counterpart of a positive real number.
rectangle
A parallelogram with four right angles.
pi
The exact numberof times the diameter of circle will go around the circle, which is approximately 3.14.
radius
The distance from the center of a circle to any point on the circle.
positive number
Any number greater than zero.
natural (or counting) numbers
Numbers that are used to count objects or things.
difference
The result of subtraction problem.
nonagon
A nine-sided polygon.
negative number
Any number less than zero.
zero
Used to describe a physical distance of no magnitude or an empty set.
acute triangle
A triangle in which all angles measure less than 90 degrees.
circumference
The perimeter of a circle.
decagon
A ten-sided polygon.
equilateral polygon
A polygon in which all segments (or sides) are the same length.
minuend
The first number in a subtraction problem.
hexagon
A six-sided polygon.
factor
The numbers in a multiplication problem.
diameter
The radius of a circle times two.
heptagon
A seven-sided polygon.
signed numbers
Numbers designated as either negative or positive by prefixing the number with either a (-) or a (+).
concave polygon
A polygon with an indentation (or cave).
square
A rhombus with four right angles.
irrational number
A number with an infinite number of digits after the decimal point.
rhombus
An equilateral parallelogram.
isosceles triangle
A triangle that has at least two sides of equal length.
positive real number
Any number that can be used to describe a physical distance greater than zero.
pentagon
A five-sided polygon.
equiangular triangle
A triangle in which the measure of all angles are equal.
quadrilateral
A four-sided polygon.
perimeter
The measure (or distance) around a polygon.
regular polygon
A polygon in which all segments have the same lenght and all angles have the same measure.
four basic math operations
Addition, subtraction, multiplication, and division
addend
Each number in an addition problem.
octagon
An eight-sided polygon
triangle
A three-sided polygon.
parallelogram
A quadrilateral that has two pairs of parallel sides.
subtrahend
The second number in a subtraction problem.
trapezoid
A quadrilateral that has exactly two parallel sides.
dodecagon
A twelve-sided polygon.
scalene triangle
A triangle in which none of the sides are equal in length.
equilateral triangle
A triangle in which the length of all sides are equal.
obtuse triangle
A triangle in which one the angles measures more than 90 degrees.
undeagon
An eleven-sided polygon.
Section 35
(52 cards)
absolute value of zero
zero
exponent
a mathematical notation indicating the number of times a quantity is multiplied by itself
dividend
The first number in a division problem.
terms
in an expression are separated by addition and subtraction signs
area of a square
A= side²
distributive property
a(b + c) = ab + ac an + ac = a(b+ c)
area
the number of square units needed to cover a flat surface
associative property of addition
changing the grouping of terms will not change the sum, (a + b) + c = a + (b + c)
formula
An equation that shows a relationship among two or more quantities
constant
a term that has no variable and does not change
commutative property for addition
The order in which two real numbers are added does not affect the sum.
perimeter
The sum of the lengths of the sides of a polygon
variable
a letter used to represent one or more numbers
evaluate
to find the value of an expression
commutative property of multiplication
the order of the factors does not change the product a x b = b x a
integers
Includes negative numbers with a set of natural numbers.
area of a circle
A = πr²
area of a rectangle
A= length x width
fraction
A division problem represent as such: 3/4.
whole numbers
Includes the number zero with a set of natural numbers.
algebraic subtraction
If a and b are real numbers, then a - b = a +(-b) where -b is the opposite of b.
order of operations
the order in which operations in an expression to be evaluated are carried out. 1. parentheses 2. exponets 3. multiplication and divison 4. addition and subtraction
area of a triangle
A= 1/2bh
mean
the sum of the values in a data set divided by the number of values in the set
coefficient
number in front of a variable
product
The result of a multiplication problem.
circumference
The distance around a circle
base
is the number that is repeatedly multiplied in a power
multiplication identity property
The product of any number and one is that number.
quotient
The result of a division problem.
solving an equation
finding all the solutions of an equation
like terms
terms that have identical variable parts raised to the same power
unit multipliers
Fractions used to change the units of a number.
absolute value
In reference to a number, the positive number that describes the distance on a number line of the graph of the number from the origin.
additive identity property
The sum of a number and zero is always that number.
additive inverse property
The sum of a number and its opposite is zero.
associative property of multiplication
changing the grouping of factors will not change the product, (ab)c = a(bc)
opposite of a number
on a number line, a number that is the same distance away from zero in the other direction
equation
a mathematical sentence with an equal sign that shows that two expressions are equivalent
numerical expression
consists of numbers and operations
set
Designates a well-defined collection of numbers.
power
is a number made of repeated factors
numerator
The number that is on top in a fraction.
denominator
The number that is on the bottom in a fraction.
commutative property of addition
In a sum, you can add terms in any order, a + b = b + a
variable expression
consists of numbers, variables, and operations
divisor
The second number in a division problem.
absolute value
the distance a number is from 0 on the number line, value of n is written l n l
integers
Positive and Negative Whole Numbers and 0. Examples: .... -3, -2, -1, 0, 1, 2, 3.....(Does not include fractions or decimals)
square unit
a square with sides one unit long.
Section 36
(50 cards)
Isolate the variable
To get a variable alone on one side of an equation or inequality in order to solve the equation or inequality
rate
a ratio that compares two quantities measured in different units
Equivalent Inequalities
inequalities that have the same solution
diameter
the distance across a circle through its center (is also twice the radius)
principal
the original amount of money loaned or borrowed
range
is the difference of the greatest value and the least value in a set of data.
radical expression
an expression that contains a square root
prime factorization
a number written as the product of its prime factors
theoretical probability
what should occur in a probability experiment...an experiment is not actually done
Equivalent Equations
equations that have the same solution
height
the perpendicular (90 degrees) distance from the base of a triangle to the opposite vertex
greatest common factor
The largest factor that two or more numbers have in common.
equivalent ratios
Ratios that have the same value.
median
is the middle value in a data set where the numbers are ordered least to greatest
ratio
a comparison of two numbers by division
quadrant
one of four sections into which the coordinate plane is divided
opposite
the number that is on the other side of 0 and is exactly the same distance away from 0
Inequality
A statement that compares two quantities using <, >, ≤,≥, or ≠
radius
a line segment from the center of a circle to any point on the circle (is also half the diameter)
interest
is the amount paid for borrowing or lending money
prime number
A whole number that has exactly two factors, 1 and itself.
Pythagorean Theorem
a² + b² = c²
reciprocal
one of a pair of numbers whose product is 1: the reciprocal of 2/3 is 3/2
equivalent fractions
fractions that have the same value and the same simplest form
base
the bottom of a triangle
Inverse Operation
operations that undo each other, such as addition and subtraction
coordinate plane
A plane that is divided into four regions by a horizontal line called the x-axis and a vertical line called the y-axis. aka "the Cartesian plane" after René Descartes
rational number
A number that can be written as a/b where a and b are integers, but b is not equal to 0.
center
the point in the exact middle of a circle
ordered pair
A pair of numbers, (x, y), that indicate the position of a point on a coordinate plane.
composite
a number that has more than two factors
multiple
skip-counting by any given number Ex: 4, 8, 12, 16...
circle
is the set of all points that are an equal distance from a point called the center
least common denominator
The least common multiple of the denominators of two or more fractions.
irrational number
a number that can not be written a/b
scale model
a model of an object in which the dimensions are in proportion to the actual dimensions of the object.
unit rate
a rate that has a denominator of 1
monomial
a single term made up of numbers and variables
circumference
The distance around a circle.
probability of an event
number of favorable outcomes divided by total number of possible outcomes
experimental probability
probability based on what happens when an experiment is actually done
mode
is the value in a data set that occurs most often. If all values occur the same amount of times there is no mode for that set.
least common multiple
the smallest multiple that two or more numbers have in common
repeating decimal
a decimal in which one or more digits repeat infinitely
proportion
an equation that states that two ratios are equal Ex: 1/2 = x/10
simplest form
when the GCF of the numerator and denominator is 1
scientific notation
a method of writing very large or very small numbers by using powers of 10
percent
a ratio whose denominator is 100
annual interest rate
apr, the percent of the principal you pay or earn a year
perfect square
a number that has a whole number for a square root ex: 9, 16, 121
Section 37
(53 cards)
Variable
A letter or symbol used to represent a number such as x, y, a, b
Pi
3.1415.... is an irrational number resulting from the ratio of a circle's circumference to its diameter
Symmetric Property
If a = b then b = a
Commutative Property of Multiplication
(x)(2) = 2x
function
a relation that assigns exactly one output value for each input value
Distributive Property
a(b + c) = ab + ac
domain
the set of all the input (x-values) for a function
Integers
{...-2, -1, 0, 1, 2...}
Multiplication Property of Zero
a * 0 = 0
Identity Property of Multiplication
1a = a
Counting Numbers
{1, 2, 3...}
Coefficient
The 2 in the term 2x
Whole Numbers
{0, 1, 2...}
Like Terms
Terms that have the same variable and the same exponent.
linear equation
an equation whose answers are ordered pairs (x,y) and whose answers graph a straight line
Associative Property of Multiplication
(ab)c = a(bc)
Term
A number, a variable, or the product of a number and a variable
independent event (probability)
an event that is not affected by another event.
Commutative Property of Multiplication
ab = ba
input
the x-value in a function
Irrational Numbers
Any number that can't be written as a fraction.
Inverse Property of Addition
a + (-a) = 0
tree diagram (probability)
a diagram used to show the total number of possible outcomes in an experiment
Inverse Property of Multiplication
a(1/a) = 1
cube
special polyhedron with faces that are all squares
Equation
2x + 3 = 8
Commutative Property of Addition
3x + 2 + 5x = 3x + 5x +2
Identity Property of Addition
a + 0 = a
permutations
...
combinations
...
fundamental counting principle
...
slope
the steepness of a line on a graph, rise over run
Rational Numbers
Any number that can be expressed as a fraction.
volume
the amount of 3-dimensional space occupied by an object
Constant
A term without a variable
x-intercept
the point where a graph crosses the x-axis
range
the set of all the output (y-values) for a function
odds
the ratio of the number of ways the event can occur to the number of ways the event cannot occur. Favorable over Unfavorable.
Associative Property of Addition
(a + b) + c = a + (b + c)
Constant
The 3 in the expression 2x + 3
Commutative Property of Addition
a + b = b + a
Associative Property of Addition
(27 + 38) + 12 = 27 + (38 +12)
Expression
2x + 3
output
the y-value in a function
Multiplication Property of -1
-1 * a = -a
y-intercept
the point where a graph crosses the y-axis
slope intercept form
y = mx + b
Pythagorean triple
a set of three positive integers that work in the pythagorean theorem
surface area
The total area of the 2-dimensional surfaces that make up a 3-dimensional object.
dependent event (probability)
an event who's outcome does depend on the outcome of a previous event
Section 38
(50 cards)
numeral
An expression that names a particular number; a numerical expression.
Addition Property of Equality
If a = b then a + c = b + c
When there are no grouping symbols, simplify in the following order:
1. Do all multiplications and divisions in order from left to right. 2. Do all additions and subtractions in order from left to right.
Multiplication Property of Zero
0x = 0
values of a variable
The numbers that can be represented by the variable.
Lesson 1-3
Equations
Additive Inverse
also known as the opposite
Identity Property of Addition
4x + 0 = 4x
Absolute Value
The distance away from zero. It is always positive.
Distributive Property
2(x + 5) = 2x + 10
equation
A statement formed by placing an equals sign between two numerical or variable expressions.
Title
Algebra Structure and Method Book 1
Subtraction Property of Equality
If a = b then a - c = b - c
Objective 1-1
To simplify numerical expressions and evaluate algebraic expressions.
Multiplication Property of Equality
x/5 = 4 so x = 20
Quotient
The answer to a division problem.
Lesson 1-1
Variables
Addition Property of Equality
3x - 2 = 7 so 3x = 9
Lesson 1-2
Grouping Symbols
open sentence
A sentence containing one or more variables.
variable
A symbol used to represent one or more numbers.
substitution principle
An expression may be replaced by another expression that has the same value.
Objective 1-2
To simplify expressions with and without grouping symbols.
Division Property of Equality
4x = 12 so x = 3
simplifying a numerical expression
Replacing the expression by the simplest name for its value.
grouping symbol
A device used to enclose an expression that should be simplified before other operations are performed. Examples: parentheses, brackets, fraction bar.
Multiplication Property of -1
-5 = (-1)(5)
Identity Property of Multiplication
1x = x
Symmetric Property
5 = x so x = 5
Sum
The answer to an addition problem.
Chapter 1
Introduction to Algebra
Division Property of Equality
If a = b then a/c = b/c
Objective 1-3
To find solution sets of equations over a given domain.
evaluating a variable expression
Replacing each variable in the expression by a given value and simplifying the result.
Reciprocal
2/3 and 3/2 8 and 1/8
Multiplicative Inverse
also known as the reciprocal
Associative Property of Multiplication
[(25)(16)](4) = (25)[(4)(16)]
Multiplication Property of Equality
If a = b then ac = bc
numerical expression
An expression that names a particular number; a numeral.
Opposites
4 and -4 -6 and 6
variable expression
An expression that contains a variable.
Subtraction Property of Equality
2x + 3 = 7 so 2x = 4
Difference
The answer to a subtraction problem.
value of a numerical expression
The number named by the expression.
Inverse Property of Addition
8 + (-8) = 0
sides of an equation
The two expressions joined by the equals sign.
Product
The answer to a multiplication problem.
Chapter 1 Section 1
Variables and Equations
Distributive Property
4x + 6 = 2(2x + 3)
Inverse Property of Multiplication
8(1/8) = 1
Section 39
(52 cards)
graph of a number
The point on a number line that is paired with the number.
Lesson 1-7
A Problem Solving Plan
Zero is niether
positive nor negative.
D = rt
Distance traveled = rate × time traveled
Objective 1-8
To graph real numbers on a number line and to compare real numbers.
solution set of an open sentence
The set of all solutions of the sentence.
negative side
On a horizontal number line, the side to the left of the origin.
satisfy an open sentence
Any solution of the sentence satisfies the sentence.
positive side
On a horizontal number line, the side to the right of the origin.
Objective 1-7
To use the five-step plan to solve word problems over a given domain.
P = 2l + 2w
Perimeter of rectangle = ( 2 × length ) + ( 2 × width )
Plan for Solving a Word Problem Step 4
Solve the equation and find the unknowns asked for.
1. If a is positive,
then −a is negative.
inequality symbols
Symbols used to show the order of two real numbers. The symbol ≠ means "is not equal to."
A = lw
Area of rectangle = length of rectangle × width of rectangle
Objective 1-5
To translate word sentences into equations.
area
The area of a region is the number of square units it contains.
real number
Any number that is either positive, negative, or zero.
origin
The zero point on a number line. The intersection of the axes on a coordinate plane.
Step 3
Reread the problem and write an equation.
Lesson 1-9
Opposites and Absolute Values
negative number
A number paired with a point on the negative side of a number line.
positive number
A number paired with a point on the positive side of a number line.
positive integers
The numbers 1, 2, 3, 4, and so on.
domain of a variable
The given set of numbers that the variable may represent.
Objective 1-6
To translate simple word problems into equations.
C = np
Cost = number of items × price per item
formula
An equation that states a rule about a relationship.
Plan for Solving a Word Problem Step 1
Read the problem carefully. Decide what unknown numbers are asked for and what facts are known. Making a sketch may help.
integers
The set consisting of the positive integers, the negative integers, and zero.
solution of a sentence
Any value of a variable that turns an open sentence into a true statement.
Step 2
Choose a variable and represent the unknows.
Lesson 1-8
Number Lines
solve an open sentence
To find the solution set of the sentence.
Plan for Solving a Word Problem Step 5
Check your results with the words of the problem. Give the answer.
coordinate of a point
The number paired with that point on a number line.
negative integers
The numbers −1, −2, −3, −4, and so on.
Lesson 1-4
Translating Words into Symbols
Lesson 1-5
Translating Sentences into Equations
opposite of a number
Each of the numbers in a pair such as 6 and −6 or −2.5 and 2.5. Also called additive inverse.
Plan for Solving a Word Problem Step 2
Choose a variable and use it with the given facts to represent the unknowns described in the problem.
Plan for Solving a Word Problem Step 3
Reread the problem and write an equation that represents relationships among the numbers in the problem.
whole numbers
The set consisting of zero and all the positive integers.
Step 1
Read the problem carefully.
Objective 1-9
To use opposites and absolute values.
Objective 1-4
To translate phrases into variable expressions.
Lesson 1-6
Translating Problems into Equations
Chapter 1 Section 3
Numbers on a Line
Chapter 1 Section 2
Applications and Problem Solving
perimeter
The perimeter of a plane figure is the distance around it.
Section 40
(53 cards)