Section 1
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Cards (182)
Section 1
(50 cards)
Rectangle
A=bh
Perpendicular Lines
-have slopes that are opposite and reciprocal
Discriminant
-for quadratic expression, the discriminant is b^2-4ac -determines tht nature of the roots when the quadratic equation=0 -if discriminant is greater than 0, 2 real roots -if discriminant=0, 1 real root -if discriminant is less than 0, 2 imaginary roots
Finding domain for radical functions
-the cadicand must never become negative so find domain by solving radicand> or equal to 0
Factoring Cubes
a^3+b^3=(a+b) (a^2-ab+b^2) -if it is A^3-B^3 then change to (a-b)(a^2+ab+b^2)
Given roots to quadratic equation and have to find the original equation
-formula is X^2-(sum of roots)x+(product of roots)=0
Circle with center at origin
X^2+Y^2=R^2
Circle
A=@r^2 @=pie C=2(pie)r or (pie)(diameter)
Vertical Line test
a relation is a function and passes the vertical line test if when a vertical line is drawn through the graph, it touches the graph only ONE time.
Distance
D=(rate)(time)
Constant of proportionality with K
look at other slides
Linear
f(x)=ax+b
relation
- a set of ordered pairs
Difference Quotient
-gives the secant slope to a curve f(x+h)-f(x)/h
Right Circular Cone Formula
V=1/3πr^2h SA=πr√(r^2+h^2 )+πr^2
function
- a relation in which each input has a unique output
Quadratic
f(x)=ax^2+bx+c
Parallelogram
A=bh
Discriminant with x intercepts
-greater than 0, 2 x intercepts -equal to 0, the vertex lies on the x axis -less than 0, the parabola doesn't intersect the x axis
Y=Kx+b
-y varies linearly as x
standard form
ax+by=c
Constant
f(x)=a
Sphere
V=(4/3)(pie)r^3 S.A.=4(pie)r^2
Finding Domain for a rational function
-omit numbers that make the denominator zero
point slope form
y-y1=m(x-x1) -passes through point (x1,y1)
Quartic
f(x)=ax^4+bx^3+cx^2+dx+e
Prisms
V=(area of base)(height)
Parallel Lines
-have equal slopes
Quadratic Formula
-b±[√b²-4ac]/2a
Right circular cylinder
V=(pie)r^2h S.A.=2(pie)r^2+2(pie)RH
Distance Formula
d = √[( x₂ - x₁)² + (y₂ - y₁)²]
Trapezoid
A=(1/2)(B+b)(H)
Irrational Number
-real number that cannot be written as a terminating or repeating decimal (square root of 5)
Position Function
-16t^2+v0t+h0 V0=initial velocity H0=initial hieght
Y=kxz
-y varies jointly as x and z
Pryamids
V=(1/3)(area of base)(height)
Circle with center at (h,k)
(X-h)^2 +(y-k)^2=r^2 -if a line is tangent to a circle, the line is perpendicular to the radius drawn to the point of tangency
Y=Kx/z
-y varies directly as x and inversely as z
Rational Number
-any number that is a real number that can be written ass a terminating or repeating decimal
Cubic
f(x)=ax^3+bx^2+cx+d
Cube
V=(edge)^3 S.A.= 6(edge)^2
Rectangular Prism
V=(lenght)(width)(height) S.A.=2(lenght)(width)+2(lenght)(height)+2(width)(height)
Midpoint Formula
(x₁+x₂)/2, (y₁+y₂)/2 -is coordinate set
Y=Kx
-y varies as x -y varies directly as x -y is directly proportional to x
Triangle
A=(B)(H)/2 - formula is one half base times hight
range
-the set of corresponding output values
domain
-the set of input values
Y=K/x
-y is inversely proportional to x
Slope intercept form
y=mx+b -m=slope -b=y intercept
zeros
-the input values that make the output zero -values for which f(x)=0 -located on the x axis
Section 2
(50 cards)
Odd functions
f(-x)=-f(x)
Parabolas
-set of all points P in a plane such that the distance from P to a fixed point F(the focus) is equal to the distance from P to a fixed line D( the directrix) y-k=a(x-h)^2 or x-h=a(y-k)^2 -has a vertex (h,k) a=1/4c or c=1/4a -absolute value of c is the distance from vertex to focus or vertex to directrix
F(x+c)
horizontal shift against the sign
intermediate value theorem
-since polynomial functions are continuous, if f(a) and f(b) are of opposite sign, there is at least one zero of f between a and b
Circle formula
(x-h)^2+(y-k)^2=r^2 -center is (h,k) -radius is r
(f+g)(x)
F(x)+g(x)
Sin&
=y/r
Finding inverse of a functions
switch x and y and solve for the new y
i
square root of negative one
Basic shapes
look at next ones
Angel measurements
angels can be measured in degrees, minutes, seconds or decimal degrees -1 degree is equal to 60 minutes (60') -1 minute is equal to 60 seconds (60'')
Decreasing
opposite -graph falls to the right
Chapter 4.2 need help
need blakes help -use synthetic division ex: x-1 put opposite of c(1) in box. Then take coefficients of the formula and put them up. Pull first one down and then multiply across.
y=square root of A-x^2
semicircle on top half
Coterminal angles
-have same initial and terminal sides and look identical -find by adding or subtracting 360 degrees, 2pie radians or 1 revolution
Polynomials
-continous smooth functions having no corners or gaps -no horizontal segments -if the degree is n, the graph has at most n x axis -maximum number of turning points is n-1 -polynomial of degree n has at most n-1 local extrema -at zero of even multiplicity, the graph touches but doesn't cross the x axis -at zero of odd multiplicity, the graph crosses the x axis -if degree is even, the graph ends in the same direction -if the degree is odd, the graph ends in opposite directions -if the leading coeffienct is positive, the graph ends in quadrant 1 -if the leading coefficient is negative, the graph ends in quadrant 4
Increasing Function
A function whose output value increases as its input value increases. -graph is increasing between (a,b) when F(b)>F(a) when b>a -graph rises to the right
Y=1/x
don't even know how to explain
f(Bx)
-horizontal stretch or squeeze
constant
as move from a to b, f(b)=f(a) -graph is horizontal
Ellipses
-set of points(x,y) in a plane, the sum of whose distances from two distinct fixed points(foci) is constant (x-h)^2/a^2 + (y-k)^2/b^2 =1 -can swap the numerators -has center (h,k) -eccentricity=c/a a^2+b^2=C^2 -c gives the distance to the focus from the center -eccentricity is less than one
(f-g)(x)
F(x)-g(x)
absolute value of f(x)
-keep all positive parts of graph -flip negative parts up
y=square root of x
is a line that comes from the x axis that extends rightward. Gets less steep as it moves along
rationalizing i in the numerator
1/(a+bi) -multiply by (a-bi)/(a-bi)
Y=x^3
flip the left side of the x^2 side down
Invertable functions
finverse{f(x)}=x
-f(x)
-reflect graph across x axis
Hyperbolas
-set of all points(x,y) in a plane, the difference of whose distances from two fixed distinct points (foci) is a positive constant (x-h)^2/a^2 - (y-k)^2/b^2 = 1 -can switch the numerators -has center (h,k) c^2=A^2+b^2 -c gives the distance from the focus to the center -asymptotes pass through (h,k) and have slope (+/-) b/a or (+/-) a/b -equation is y-k=(+/-)b/a(x-h) -eccentricity is greater than 1
f(x)+d
-vertical shift with the sign
(f/g)(x)
F(x)/G(x) -g(x) can't equal zero
Af(X)
-vertical stretch or squeeze -multiply each y value by A
i^4
1
intercept form for a parabola
y=a(x-r1)(x-r2) a gives shape r1 and r2 are the x intercepts vertex x=(r1+r2)/2 y=f((r1+r2)/2)
chapter 4.8 need blakes help
need blakes help
positive and negative angles
-positive angles are measured counterclockwise -negative angles are measured clockwise
discriminant with parabola
b^2-4ac >0 then the parabola has 2 x intercepts <0 parabola has 0 x intercepts =0 the parabola has a vertex on the x axis
Y={x}
-don't even know
Y=x^2
Is a parabala im pretty sure. Looks like the square root of x one but intersects at the origin and has two sides
i^2
-1
i^3
-i
Cos&
=x/r
parabola standard form
y=ax^2+bx+c -a gives shape =1 is normal, >1 is fat,<1 is skinny(absolute values) -- if a >0 opens up, <0 opens down -c is the y intercept -b gives movement on the horizontal axis find the vertex by x=-b/2a, y=f(-b/2a)
Converting between radians and degrees
multiply by (pie)/180 degrees or 180/(pie)
Even functions
f(-x)=f(x)
f(-x)
-reflect graph across y axis
Y=absolute value of x
Looks like a V that intersects at the origin
Completed square form for a parabola
y=a(x-h)^2+k a gives shape (h,k) is the vertex
f(absolute value of x)
-keep positive x function value -mirror those values across the y axis
(F*g)(x)
F(x)*G(x)
Section 3
(50 cards)
cotx
Domain- x cannot equal n(pie) Range-all reals Asymptotes = what domain cannot be
tan inverse
domain (- infinity, infinity) Range (-Pie/2,pie/2)
csc(x)
Domain- same as cot Range- same as sec(x) Asymptotes- same as cot(x)
Solving trig equations
sine is positive choose from quadrants 1 and 2 cosine is positive choose from quadrant 1 and 4 tangent is positive, choose between quadrant 1 and 3
b^log(x)
=x
Natural Logarithms
ln(x) refers to loge(x) where e=2.718
tan x
Domain x cannot equal (2n+1)(pie)/2 Range- all reals Asymptotes= what domain cannot equal
logb1
=0
Value of investment that grows continuously
A=Pe^rt
sec&
1/cos
quantity is decaying at a rate of r
f(x)=a(1-r)^t
half angle formulas
chapter 8.3
linear velocity
v=s/t v=r&/t -measure of how fast a point moves around the circle
cos(x)
Domain- all reals range [-1,1] Asymptotes= none
cos inverse
Domain[-1,1] Range[0,pie]
Sum and difference formulas
sin[x(+/-)y]=sin(x)cos(y)[+/-]cos(x)sin(y) cos[x(+/-)y]=cos(x)cos(y)[-/+]sin(x)sin(y) ----notice that for cos, if it is positive in the first part, is negative in the second part tan[x(+/-)y]=tan(x)[+/-]tan(y)/1[-/+]tan(x)tan(y) ---notice that the top and bottom are opposite of each other
Pythagorean identities
sin^2+cos^2=1 tan^2+1=sec^2 1+cot^2=csc^2
Finding the value of an investment
A=P(1+r/n)^nt -P=amount invested -r=annual rate -n=number of times compunded per year -t=number of years
sec(x)
Domain-same as tangent Range=(-infinity,1] in union with [1,infinity) asymptotes= same as tangent
Following rules hold for x,y>0 but don't equal 1
Following rules hold for x,y>0 but don't equal 1
logb(a)
log(a)/log(b) = ln(a)/ln(b)
arc length
s=r&
angular velocity
w=&/t -is the measure of how fast the central angle is changing
exponential functions
f(x)=a*b^x -a and b are constant and the variable is the exponent -A gives the initial value -if B>1 then we have exponential growth -if B<1 then we have exponential decay
Characteristics of even, power functions
f(-x)=f(x) -graph has even or y axis symmetry -domain is all reals -range is nonnegative real numbers -graph always contains (0,0), (-1,1) and (1,1) -as the exponent grows in magnitude, the graph flatens toward the origin and becomes steeper as the absolute value of x grows -both ends go in the same direction
cot&
1/tan
normal period for tangent and cotangent
pie
Cot&
cos&/sin&
Csc&
=1/sin
ln(x)=k
=loge(x)=k so e^k=x
area of sector
A=(1/2)r^2
Double angle formulas
sin(2x)=2sin(x)cos(x) cos(2x)=cos^2-sin^2 =1-2sin^2 =2cos^2-1 tan(2x)=2tanx/1-tan^2
logb(x^k)
=k*logb(x)
Sin(x)
Domain-all reals Range [-1,1] Asymptotes= none
Negative angle formulas
sin(-x)=-sin(x) cos(-x)=-cos(x) tan(-x)=-tan(x)
logb (xy)
=logb(x)+logb(y)
Tan&
=y/x
LogaB=k
a^k=b
If quantity is growing at a rate of r
f(x)=a(1+r)^t
logb(x^b)
=x
Complementary angle identities
sin[(pie/2) -&]=cos& tan[(pie/2) -&]=cot& sec[(pie/2)-&]=csc&
relating linear and angular velocity
v=rw
logb(x/y)
=logb(x)-logb(y)
normal period for sin, cosine, secant and cosecants
2(pie)
in unit circle
r=1
graphing y=Asin[B(x-c)]+D
-absolute value of A is the amplitude. if less than 0, the graph flips -B produces a period change( the new period is the normal period divided by B) -C produces a horizontal shift. Sin(x-pie) moves to the right pie units -D is the vertical shift. goes with sign
common logs
log(x)=log10(x)
sin inverse
domain[-1,1] range [-(pie)/2,(pie)/2]
Characteristics of odd, power functions
f(-x)=-f(x) -graph has odd or origin symmetry -domain and range are all reals -graph contains (0,0), (-1,-1), and (1,1) -as the exponent increases in magnitude, the graph flattens toward the origin and becomes steeper as the absolute value of x increases -the ends go in opposite directions
tan&
sin&/cos&
Section 4
(32 cards)