Section 1
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Cards (356)
Section 1
(50 cards)
Initial Side
the fixed ray of an angle
Sine, Cosine, Tangent on the unit circle
Periodic Function
a function with a repeating graph
Quadrant Angle
An angle whose terminal side is on an axis
Phase Shift
A horizontal translation of a periodic function
Frequency
the reciprocal of the period
Radian
a unit of angle, equal to an angle at the center of a circle whose arc is equal in length to the radius
Cosine
cos(x)=adj/hyp
Sector
The part of a circle that looks like a piece of pie. A sector is bounded by 2 radii and an arc of the circle
Cotangent
cot(x)=adj/opp
Cosecant
csc(x)=hyp/opp
Amplitude
half of the difference of the maximum value and the minimum value
Finding terms in arithmetic sequence
an = a1 + (n - 1)d
Conterminal Angles
Coterminal angles are just different ways of naming the same angle.
Quadratic formula (Find the Roots)
the quadratic formula is x equals negative b plus or minus the square root of b squared minus 4 times a times c all over 2 times a.
Midline
the point in between the maximum and minimum points (ex. 0 would be the midline)
Canceling out opposites
If both terms are opposite signs they can be crossed out and replaced by a negative 1.
Central Angle
θ of a sector is the angle formed by the two radii
Common Logarithms
Convert degrees to radians
To convert from degrees to radians multiply the degrees by pi and divide by 180.
Exact values of trigonometric functions
Natural Logarithms
Terminal Side
a ray of an angle that rotates about the center
Square root equations
To solve square root equations Isolate the square root on one side of the equation. Square both sides of the equation Check the answer(s) in the original equation for extraneous roots.
Polynomial
•An algebraic expression with one or more terms •The prefix POLY means MANY
Cycle
The shortest repeating portion of the graph
Period
the horizontal length of each cycle
Factoring out a -1
To factor out a -1, remove the negative and change the signs of the terms
Converting between logarithm and exponent form
Snail it!
Cotangent
cotangent equals 1 divided by tangent or cosine divided by sine.
One to one function
A one to one function must pass the vertical and horizontal line test. No x or y coordinate can repeat.
Complex fractions
Unit Circle
a circle with a radius of 1, centered at the origin
Rationalizing the denominator
To rationalize a denominator multiply the numerator (top) and denominator (bottom) by the conjugate of the denominator (bottom).
Coterminal
when two angles have the same initial and terminal sides
Logarithms
Domain, range, inverse
The domain is the x coordinates. The range is the y coordinates. To find the inverse switch the x and y coordinates.
Raising a negative number to a power
When raising a negative number to a power always put the negative number in a parenthesis.
Secant
sec(x)=hyp/adj
Factoring by grouping
Sine
sin(x)=opp/hyp
Equations with fractions
To solve equations with fraction, find a common denominator. Drop the denominators Solve the resulting equation. Check the answer(s) in the denominators for extraneous roots.
Tangent
Tangent equals sine divided by cosine.
Degrees and radians
Once around a circle is 2 pi radians or 360 degrees. ½ of the way around a circle is pi radians or 180 degrees. ¼ of the way around a circle is pi divided by 2 or 90 degrees.
tangent
tan(x)=opp/adj
Referance Angle
the acute angle formed by the terminal side of and the x-axis
Conjugate
To find the conjugate The first term stays the same. Change the sign of the second term.
Convert radians to degrees
To convert from radians to degrees, multiply the radians by 180 and divide by pi.
Standard Position
When the vertex is at the origin and one ray is on the positive x-axis
Operations with fractions
Section 2
(50 cards)
Range
All possible values of the dependent variable(output)
Natural numbers
Counting numbers greater than 0
Function
A relation in which each element of the domain is paired with exactly one element of range. (If DOMAIN is chosen more than once, it is NOT A FUNCTION) •In a function, a vertical line intersects the graph only ONCE(The vertical line test)
The ___________________ depends on the ______________________
Dependent variable(output), independent variable(input)
!Checklist for graphing a line from an equation!
•Label x and y axis •Connect at least 2 points with a STRAIGHT EDGE •Draw arrows at both ends of the line •Label the line with the equation
Algebraic equation
Any sentence with an equal sign
Quadratic
Linear system is made up of a quadratic equation (palabra) and a linear equation
Special line: Vertical
If a line crosses X-AXIS we write x=?
To factor difference of perfect squares, REMEMBER:!
•The numerical coefficient has to be a perfect square •And the exponent of each variable has to be an EVEN NUMBER.
Dinding the _______ is the same as finding the ______________________
Slope, average rate of change
Special line: Horizontal
If a line crosses Y_AXIS, we write y=?
Domain
All possible values of the independent variable(input)
Lines intersected
One solution, different slopes, different y-intercepts
Standard form of quadratic function
y=ax^2+bx+c
Parallel line
Do NOT intersect; they have the same slope but DIFFERENT Y-INTERCEPTS.
Lines coincide
Infinitely many solutions; same y-intercept; same slope
Linear equation
An equation whose graph in a line. The points on the line are SOLUTIONS of the equation.
Function notation
y=f(x) y=output f=Name of function x=input
Quadratic equation
•ax^2+bx+c=0 •The solution(s) to this are called ROOTS or ZEROS of the function (solution are x-intercept)
To complete the square
1)Find one half of b-> b/2 2)Square what you got from step 1 3)Add the result to the original expression
Integers
Positive and negative whole numbers
Independent variable
X
Dependent variable
Y
Substitution Method
1)Isolate a variable in at least one of the equations 2)Substitute for this variable into the other equation 3)Solve the new equation 4)Substitute the result into one of the equations to solve for the other variable 5)Check in each equation
Number sentence
A specific type of algebraic equation. It is an algebraic expression that only has numbers (NO VARIABLES)
Multiplying rational expressions
1)Factor 2)Cancel all common factors in the numerator and denominator 3)Multiply straight across
Elimination Method
1)Line up the variables 2)Find the variable with opposite coefficient 3)Add the equations to ELIMINATE the variable 4)Solve the new equation 5)Find the other variable by substitution 6)Check!
Rational numbers
Can be written as a fraction or a decimal which is repeating
Cubic Functions
•Are functions with a degree of 3 •Standard form= ax^3+bx^2+cx+d (a CANNOT equal 0)
To solve quadratic equation
•Rewrite equation in standard form •Factor •Set each factor equal to 0 •Check each solution with equation
The axis of symmetry
The vertical line that cuts a function down the middle.
Inequality
A mathematical statement that contains an inequality symbol (!!! Whenever you MULTIPLY or DIVIDE an inequality by a NEGATIVE NUMBER, you MUST FLIP the inequality sign!!!)
Lines are parallel
No solution, same slopes, different y-intercepts
Solid line
<= and >=
To simplify radicals
•Find the largest perfect square which is a factor of the radicand •Rewrite the square root as a product of a perfect square and another factor •Evaluate the square root of the perfect square and write as a coefficient •Keep the other factor as a radicand. Ex: √50= √25 √2 => 5√2
Step function
A piece-wise function defined by CONSTANT values over its domain. The graph of a step function consists of a series of line segments.
Algebraic expression
A mathematical phrase that can include numbers, variables, and operation symbols
Dashed line
< and >
Dividing rational expressions
1)Keep the first term 2)Change division to multiplication 3)Flip second term (to get reciprocal)
To factor COMPLETELY
•Check for a GCF and factor it out if possible •Factor further if possible (different of perfect squares or a quadratic trinomial) •Check by multiplying using the distribution property
When a<o
Opens downwards :(
When a>0
Opens upward :)
Whole numbers
Natural numbers including 0
Irrational numbers
Cannot be expressed as a fraction, they are non-repeating or non-terminating decimals.
Finding the axis of symmetry & vertex
x=-b/2x
Factoring y=ax^2+bx+c the x-box way
1)Find a*c (product of first and last term) 2)Find 2 new factors 3)Fill in the box
Quadratic formula
x=-b+-√b^2-4ac/2a
Perimeter
2L+2W
Steps to simplify rational expressions
1)Factor what is possible in numerator & denominator 2)Reduce!
Variable
A symbol (usually a letter) representing one or more unknown numbers
Section 3
(50 cards)
Subtraction Property of Equality
If a=b, then a-c=b-c
Base
the value that is raised to a power when a number is written in exponential notation. in the term 5³, 5 is the __________.
1 hour= .... minutes
60
Coefficient
Number in front of a variable
Inverse
Opposite
Division Property of Equality
If a=b, then a / c=b / c
In the formula for Arithmetic Sequences, an=a1+(n-1)d, what does n, a1, and d stand for?
n= the number of the term you are looking for A1= the first term D= the common difference
Rational + Rational=
Rational
Rational + Irrational=
Irrational
In the formula A=P(1+r)^n what does P, r, and n stand for
P= Original Amount r= Rate as a decimal n= Number of time periods over which the value takes place
Equation
A mathematical statement that two expressions are equal.
1 week= .... days
7
What is a polynomial?
An equation with more than one term
What is a monomial?
A equation with one term
What is the formula for exponential functions? (DECREASE)
A=P(1-r)^n
Term
the parts that are added together 2x and 3 are terms
Simplify
to write an expression in a simpler form
Commutative
8 + 3 = 3 + 8
What is a positive correlation in a scatterplot?
Points resemble a straight line with a positive slope (r close to +1)
What is the formula for slope?
y^2-y^1 -------- x^2-x^1
1 year = ... weeks
52
Distributive Property
6(3 + 5) = (6 x 3) + (6 x 5)
1 minute= .... seconds
60
For a box plot, you would need: ......
-Lower Extreme (Min X) -Lower Quartile (Q1) - Median (Med) - Upper Quartile (Q3) - Upper Extreme (Max X)
In a graph, functions need to pass the ......
Vertical Line Test
The negative exponent rule
1)Move the base of the negative exponent to the opposite part of the fraction 2)Change the sign from negative to positive.
In a piecewise function, for > or <, you would use a ..... circle.
Open
Exponent
the number of times a number is multiplied by itself
Evaluate
solve or put a number into an equation
In a piecewise function, for > (underlined) and < (underlined), you would use a ..... circle
Closed
Multiplication Property of Equality
If a=b, then a c=b c
Perimeter
The distance around the outside of a shape
What is the formula for Exponential Functions? (INCREASE)
A=P(1+r)^n
What are Irrational Numbers?
non-repeating, non-terminating decimals, square roots of non-perfect squares, π and most operations with π
Solution of a system of linear inequalities
Every point in the overlapping region
Constant Term
term without a variable 3 is the _______ _______
1 year = .... days
365
What are Rational Numbers?
Repeating of terminating decimals, integers, square roots of perfect squares, functions consisting of two terms
1 yard = ... feet
3
1 yard = .... inches
36
Order Of Operations
PEMDAS or rules for solving problems Parentheses Exponent Multiply or Divide Add or Subtract
1 foot = ... inches
12
Addition Property of Equality
If a=b, then a+c=b+c
Associative
2 + ( 3 + 4 ) = (2 + 3 ) + 4
What is no correlation in a scatterplot?
Points will be scattered all around the graph (r is close to 0)
What is a negative correlation in a scatterplot?
Points resemble a straight line with a negative slope (r close to -1)
What are the Properties of Equality?
1. Addition Property of Equality 2. Subtraction Property of Equality 3. Multiplication Property of Equality 4. Division Property of Equality
If a relation is a function, ..... values CANNOT REPEAT
X
1 dozen= .... items
12
What is the formula of the axis of symmetry for a parabola?
-b x= ---- 2a
Section 4
(50 cards)
Inequality
A mathematical sentence that contains less than, greater than, less than or equal to, greater than or equal to, or not equal
Standard Form
Ax+By=C
Relation
A link between two tables of numbers based on a equation
Origin
the point (0,0) where the x-axis and the y-axis intersect in a coordinate plane
Or
The variable is in two different places
Expression
A mathematical phrase involving at least one variable and sometimes numbers and operation symbols
Complex Conjugate
A pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs. For example, 3 + 4i and 3 − 4i are complex conjugates.
Substitution Method
Replacing one variable with an equivalent expression containing the other variable
Direct Variation
y=kx
Solution
The answer to an equation
Point Slope Form
y-y1=m(x-x1)
Equivalent
Equal
Slope Intercept Form
y=mx+b
Vertical Line Test
A test used to determine whether a relation is a function by checking if a vertical line touches 2 or more points on the graph of a relation
Vertex Formula
Absolute Value Function
a function written in the form y = /x/, and the graph is always in the shape of a v
Factoring
Line of Best Fit
a line drawn in a scatter plot to fit most of the dots and shows the relationship between the two sets of data
Vertex
The corner of the absolute value function
Difference of Squares
Vertex Form
Independent Variable
A variable whose variation does not depend on that of another. x
Ordered Pair
a pair of numbers ,or coordinates,(x,y) describing the location of a point on the coordinate plane
Linear Inequality
An inequality in two variables whose graph is a region of the coordinate plane that is bounded by a line.
System of Equations
A set of equations with the same variables
Indirect Variation
y=k/x
The Discriminant
Complex number
A complex number is the sum of a real number and an imaginary number (a number whose square is a real number less than zero), i.e. an expression of the form where a and b are real numbers and i is the imaginary unit, satisfying i^2 = −1.
The Quadratic Formula
GCF
Greatest Common Factor
Domain
The input numbers of x numbers
Slope
(y₂-y₁)/(x₂-x₁)
Parallel
Slopes are the same
Constant of Variation
what the k is in the equation y = kx. Like the slope or rate of change
Variable
A letter used to represent one or more numbers
1-3-5 Rule
Shortcut for graphing a parabola
Binomial Expression
An algebraic expression with two unlike terms
Range
The output or y values
Linear Equation
A one variable equation written in the form ax=b
Solution of a System
Any ordered pair in a system that makes all the equations of that system true.
Function Notation
uses f(x) instead of y
Elimination Method
another name for combination method solving systems by adding or subtracting equations to eliminate a variable
Perpendicular
The slopes are opposite reciprocals
And
The variable is between two numbers
Dependent Variable
The output variable; the variable that may change in response to the independent variable
Reciprocal
The number flipped over into a fraction. A pair of numbers whose product is 1
Exponential functions
A function of the form y=a·bx where a > 0 and either 0 < b < 1 or b >1.
Absolute Value
The distance a number is from zero on a number line. ALWAYS POSITIVE
Parabola
A curved line on a plane
System of Inequalities
set of two or more inequalities with two or more variables
Section 5
(50 cards)
ordered pair
(x,y), a point on a graph
equivalent inequalities
inequalities that have the same solution
no solution graph
Rational exponents
For a > 0, and integers m and n, with n > 0
Real numbers
All numbers
Rational expression
A quotient of two polynomials with a non‐zero denominator
rational numbers
Any number that can be expressed as a fraction
elimination method
answer to a system of linear inequalities
the area shaded by both graphs
opposite
switching of signs
constant term
a term with no variables
The properties of operations
Here a, b and c stand for arbitrary numbers in a given number system. The properties of operations apply to the rational number system, the real number system, and the complex number system.
infinitely many solution
extraneous solution
false answer
If the exponents values are equal in the numerator and denominator then, m=n, then....
then the line has a horizontal asymptote at y=a/b where a and b are the leading coefficients.
Simplified Form
the numerator and denominator have no common factors.
Integers
Positive and negative whole numbers
Whole numbers
Natural numbers ( counting numbers) and zero; 0, 1, 2, 3...
Rational number
A number expressible in the form a/b or - a/b for some fraction a/b. The rational numbers include the integers
Linear equation
an equation whose graph is a line
formula
is an equation that relates two or more quantities
exponent
power; 3^4 it is the 4
Equation
A mathematical sentence that contains an equals sign.
linear inequality
can be formed by replacing the equal sign in a linear equation with an inequality symbol
exactly one solution
If the exponent in the denominator is larger than the exponent in the numerator; m<n, then...
the line has a horizontal asymptote at y=0.
system of two linear equations
two variables x and y also called a linear system consists of two equations
Rational Function
A function that can be written as a polynomial divided by a polynomial.
Solution
answer
Nth roots
The number that must be multiplied by itself n times to equal a given value. The nth root can be notated with radicals and indices or with rational exponents, i.e. x1/3 means the cube root of x.
variable
A letter used to represent one or more numbers
base
3^4; it is the 3
If the exponent in the numerator is larger than the exponent in the denominator; m>n, then...
the graph has no horizontal asymptote.
system of three linear equations
a system consisting of three linear equations in three variables
Irrational numbers
Numbers that cannot be written as fractions, go on forever
Trinomial
An algebraic expression with three unlike terms
Polynomial function
A polynomial function is defined as a function, f(x)=ax^n+ bx^n-1+...+z , where the coefficients are real numbers.
Whole numbers
The numbers 0, 1, 2, 3, ....
power
exponent
Standard Form of a Polynomial
To express a polynomial by putting the terms in descending exponent order.
ordered triple
(x,y,z)
system of linear inequalities
terms
the parts of an expression that are added together
Reciprocal
The multiplicative inverse of a number
compound inequality
Two or more inequalities joined together by "and" or "or"
absolute value
The distance a number is from zero on a number line
equivalent equations
equations that have the same solution
Coefficient
The number in front of a variable
to add or subtract rational exponents
the denominators need to be the same.
substitution method
Section 6
(50 cards)
composition of a function
f(g(x))
Growth Factor
the b value of a growth function, greater than 1
cube root
An idea similar to that of a square root, except that the root must be cubed
like radicals
radical expressions that have the same index and radicand
Leading Coefficeint
number in front of the highest degree
standard form
degrees decrease going left to right
rational function graph
Exponential Growth Function
A function of the form y=ab^x, where b is greater than 1 and a is greater than zero
natural log
base e or ln
cubic function
a polynomial function with the highest power being three
Power of a product
(ab)^m = a^m b^m
Exponential Equations
equations in which variable expressions occur as exponents
quadratic function
a function in which the greatest power of the variable is 2
Quotient Property of logarthims
logb (m/n) = logb m - logb n
What are the shifts of the following y=2√(x+3)-1
down 1, left 3, and stretched by 2
Exponential Function
a function of the form f(x) = ab^x, where a and b are real numbers with a ≠ 0, b>0, and b ≠ 1
cross multiplying
solve a rational equation when each side of the equation is a single rational expression.
rational exponent
fraction exponent
Domain
x-values, input
asymptote
is a line a graph approaches
Zero Exponent
a^0 = 1
polynomial function
a function that is represented by a polynomial equation
common log
base 10 log
inverse function
A relation formed by reversing x and y in a function.
index of a radical
the value of n in the radical ^n√a
Polynomial
is a monomial or the sum of monomials
Complex Fraction
is a fraction that contains a fraction in its numerator or denominator
Power of a power
(a^m)^n = a^mn
Vertical Asymptote
A vertical line that a graph approaches but never reaches.
Range
y-values, output
Power of a Quotient
(a/b)^m= (a^m)/(b^m)
Logarithmic Equations
equations that involve logarithms of variable expressions
Least common Denominator
The least common multiple of the denominators of two or more fractions.
Decay Factor
the b value of a decay function, between 0 and 1
radical function
a function that contains a radical with a variable in its radicand
Exponential Decay Function
y = ab^x if a > 0 and 0 < b < 1
product property of logarithms
logb mn = logb m + logb n
Constant term
term without a variable
(a+b)(a-b)
a^2-b^2
What are the shifts of the following y=2√(x-1)+3
up 3, right 1, and stretched by 2
Euler's Number
e
Product of Powers
a^m * a^n = a^m+n
radical sign
√
logarithm of y with base b
The expression y = logb x is a logarithmic function
Degree
largest expoenent
Quotient of powers
a^m/a^n = a^m-n
Horizontal Asymptote
a horizontal line that the curve approaches but never reaches
Power Property of logarthims
logb m^n = nlogb m
Negative exponent
a^-n = 1/a^n
What is the index of 3^(5/3)
The index is 3
Section 7
(50 cards)
binomial
two terms
(a-b)^3
a^3-3a^2b+3ab^2-b^3
Intercept Form
y=a(x-p)(x-q)
completing the square
Slope-Intercept Form
radicand
polynomial with a degree of 3
cubic
(a+b)^2
a^2+2ab+b^2
A relation for which each input has exactly one output
Function
Radical sign
√
a^2-2ab+b^2
(a-b)^2
polynomial with a degree of 1
linear
complex number
a+bi
quadratic formula
imaginary number
i
y-intercept
where the graph crosses the y-axis; x=0
Horizontal Line
slope is 0
absolute value of a complex number
√a^2+b^2
synthetic division
the shortcut for long division of polynomials when dividing divisors of the form x - k
polynomial witha degree of 0
constant
Vertex Form
y=a(x-h)^2+k
polynomial with a degree of 4
quartic
Linear function
a function in the form y=mx+b
Parallel lines
two lines who don't intersect; same slope
discriminant
b^2-4ac
roots, zeros, x-intercepts
Solutions to a quadratic equation
trinomial
3 terms
(a+b)^3
a^3+3a^2b+3ab^2+b^3
conjugate
a+bi, a-bi
m=
(y2-y1)/(x2-x1)
complex plane
place to graph complex numbers
Standard Form
Ax+By=C
square root
√a
polynomial with a degree of 2
quadratic
Long division
Long dividing equations by binomials to find the roots
Axis of Symmetry
divides the parabola into mirror images; passes though the vertex
(a-b)^2
a^2-2ab+b^2
Vertex
The turning point
f(x)=|x|
Monomial
one term
quadratic equation
ax^2+bx+c=0
Quadratic Function
y=ax^2+bx+c
Parabola
x-intercept
where the graph crosses the x-axis; y=0
slope
rate of change, rise over run
Relation
a mapping, or pairing, of input values with output values
a^2-b^2
(a-b)(a+b)
i
√-1
Perpendicular lines
Two lines who cross at 90 degrees, opposite reciprocal slope
Function notation
f(x)=mx+b
Section 8
(6 cards)