final exam Flashcards
triangle formulas
A= 1/2bh P= a+b+c
rectangle formulas
A=lw
P=2(l+w)
trapezoid
A=1/2(a+b)h
P= a+b+c+d
ellipse
A=πab
square
A=a^2
P= 4a
parallelogram
A=bh
P= 2(a+b)
circle
A=πr^2
C=2πr
sector
A=1/2r^2θ
rectangular prism
SA= 2(wh+lw+lh) or x^2+4xh
V=lwh
*Rectangular Box
cylinder
SA= 2πr^2+2πrh V= πr^2h
guidelines for modeling w functions
- express model in words
- choose the variable
- set up the model
- use model
story problem steps
- get the question
- identify the variable(s)
- build equations
- solve
- ask “does my answer make sense?”
- state answer as a sentence
transformations steps
- sketch basic graph
- do horizontal shifts
- do reflections
- do vertical shifts
inverse functions
- f(x1) ≠ f(x2)
- f(x1) = f(x2)
f^-1(y)=x; f(x)=y
cancellation props of inverse functions
if f and f^-1 are inverse functions, f^-1(f(x))=x f (f^-1(x))=x if f and f^-1 have the property *, we say f and f^-1 are inverse functions *anything >1 is undefined
technique to find inverse functions
- interchange x+y roles
- solve for y in terms of x
- set f^-1(x)=y
average rate of change
f(b)-f(a)/b-a
*diff quotient (x=a; x=a+h)
f(a+h)-f(a)/h
domains for combining functions
(f+g), (f-g), and (fg) are D=A∩B
(f/g) is D{x∈A∩B I g(x) ≠ 0}
x= b _slope; y=a _slope
und; 0
inequalities steps
- move all nonzero terms to one side
- factor
- find interval
- sign chart w test values
- solve
complete the square steps
- make sure a=1
- subtract c (x^2+bx= -c)
- find (b/2)^2
- add that value to both sides
- factor
- take square rt of both sides
- solve
a^m/n =
(^n√a)^m
a^b+c =
a^b*a^c
(a^b)^c
a^b*c
e is about
2.71828
standard form
f(x)= a(x-h)^2+k
h=
-b/2a
k=
plug h in function
if a>0, there is a
minimum value of k at x=h
if a<0, there is a
maximum value of k at x=h
graphing polynomial function steps
1) intercepts
2) sign check
3) end behavior
4) graph and label
when there is a multiplicity of 1, it
crosses the x axis
a^2+b^2=
(a+bi) (a-bi)
i^2 is
-1
use __ to determine real zeros of polynomials
long division
if c is a zero of p(x),
then x-c is a factor of p(x)
if a+bi is a zero of p(x),
a-bi is also a zero of p(x)
coefficient
the number ex: 4
term
the whole thing ex: 4x^2
domain constraints
- 1/x ;x≠0
- √x ;x≥0 *n is even
- 1/√x ;x>0 *n is even *root is alone
- logaX; x>0
graphing rational functions steps
- factor
- intercepts
- x int on numerator
- plug in 0 for y int - VA (x=)
- solve denominator
* sign chart - HA (y=)
- graph and label
logarithmic properties
loga1= 0 logaA= 1 logaA^x=x a^logaX= x logaA^c= ClogaA loga(AB) = logA+logB loga(A/B)= logA-logB
basic unit circle values
x^2+y^2=1
π/6 (√3/2, 1/2)
π/4 (√2/2, √2/2)
π/3 (1/2, √3/2)
exponential and log equations steps
1) put exponential on one side
2) solve
trig functions
sin^2(x) + cos^2(x)= 1
tan^2(x)+1=sec^2(x)
1+cot^2(x)=csc^2(x)
degrees to radians
multiply π/180
radians to degrees
multiply 180/π
coterminal angles
add or subtract 2π (360 degrees) to given angle
trig functions of angles
SOH CAH TOA
a^2+b^2=c^2
basic trig function domain and range
sin(θ)
domain: (-∞,∞)
range: [-1,1]
cos(θ)
domain: (-∞,∞)
range: [-1,1]
tan(θ)
domain: (-∞,∞) {π/2+nπ}
range: (-∞,∞)
inverse trig functions domain and range
sin-1(θ)
domain: [-1,1]
range: [-π/2, π/2]
cos-1(θ)
domain: [-1,1]
range: [0, π]
tan-1(θ)
domain: (-∞,∞)
range: (-π/2, π/2)
even-odd identities
sin(-x)= -sinx cos(-x)= cosx tan(-x)= -tanx
guidelines for proving trig identities
1) start with one side
* indicate which side u choose to start with
2) use known identities
3) covert to sines and cosines
formula for sine
sin(x+y)= sinXcosY + cosXsinY
sin (x-y)= sinXcosY - cosXsinY
formula for cosine
cos(x+y)= cosXcosY - sinXsinY cos(x-y)= cosXcosY + sinXsinY
double-angle formulas
sine: sin2x= 2sinxcosx
cosine: cos2x= cos^2x-sin^2x
= 1-2sin^2x
= 2cos^2x-1
tangent: tan2x=sin2x/cos2x
sine graph
sink(x-b)+c period: 2π/k amp: IaI phase shift: b d=period/4 *always starts with "b" *basic graph starts from 0
cosine graph
acosk(x-b)+c period: 2π/k amp: IaI phase shift: b d=period/4 *always starts with "b" *basic graph starts at highest point
tangent graph
atank(x-b)+c period: π/k d=period/4 *b in the middle *2 asymptotes at the very ends
basic trig equation steps
1) find primary solution in one complete period
sin [0,2π) cos [0,2π) tan (-π/2, π/2)
2) find general solution by adding the solution in step 1 by the multiple of the period
*sin and cos: add 2kπ
*tan: add kπ
5-step strategy
1) write down in one function of one angle
2) find values of written function
3) solve for angle
4) solve for variable
5) check restrictions
basic trig equations CHECK
1) factor
2) identities
3) formulas
- addition/subtraction
- double angle
* u substitution
if inverse function is on the outside, look at the
domain
if inverse function is on the inside, look at the
range
solving exponential/log equations: exponential
1) isolate exp
2) take loga
3) solve for variable
* no check
* sometimes we can factor
solving exponential/log equations: log
way 1) 1. isolate loga 2. write in exp form 3. solve for variable 4. check way 2) logaX=logaY; X=Y
whenever u see NONLINEAR inequalities you must solve by ___
sign chart
half angle formulas
sin θ/2 = ± √(1-cosθ)÷2
cos θ/2= ± √(1+cosθ)÷2
tan θ/2= sin θ÷(1+cosθ)
= (1-cosθ)÷sinθ
basic graph of e
above x-axis
increasing from left to right
crosses (0,1)
basic graph of log
increasing from - y values to + y values passes thru (1,0) VA x=0
a^2 x b^2 =
(ab)^2
bad point
a point on the denominator of a non-linear inequality
zero point
real zeros *set equal to zero
how do u write factors that are 1 ± 2i
(x-(1-2i)) (x-(1+2i))
