final exam Flashcards
(74 cards)
1
Q
triangle formulas
A
A= 1/2bh P= a+b+c
2
Q
rectangle formulas
A
A=lw
P=2(l+w)
3
Q
trapezoid
A
A=1/2(a+b)h
P= a+b+c+d
4
Q
ellipse
A
A=πab
5
Q
square
A
A=a^2
P= 4a
6
Q
parallelogram
A
A=bh
P= 2(a+b)
7
Q
circle
A
A=πr^2
C=2πr
8
Q
sector
A
A=1/2r^2θ
9
Q
rectangular prism
A
SA= 2(wh+lw+lh) or x^2+4xh
V=lwh
*Rectangular Box
10
Q
cylinder
A
SA= 2πr^2+2πrh V= πr^2h
11
Q
guidelines for modeling w functions
A
- express model in words
- choose the variable
- set up the model
- use model
12
Q
story problem steps
A
- get the question
- identify the variable(s)
- build equations
- solve
- ask “does my answer make sense?”
- state answer as a sentence
13
Q
transformations steps
A
- sketch basic graph
- do horizontal shifts
- do reflections
- do vertical shifts
14
Q
inverse functions
A
- f(x1) ≠ f(x2)
- f(x1) = f(x2)
f^-1(y)=x; f(x)=y
15
Q
cancellation props of inverse functions
A
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
16
Q
technique to find inverse functions
A
- interchange x+y roles
- solve for y in terms of x
- set f^-1(x)=y
17
Q
average rate of change
A
f(b)-f(a)/b-a
*diff quotient (x=a; x=a+h)
f(a+h)-f(a)/h
18
Q
domains for combining functions
A
(f+g), (f-g), and (fg) are D=A∩B
(f/g) is D{x∈A∩B I g(x) ≠ 0}
19
Q
x= b _slope; y=a _slope
A
und; 0
20
Q
inequalities steps
A
- move all nonzero terms to one side
- factor
- find interval
- sign chart w test values
- solve
21
Q
complete the square steps
A
- 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
22
Q
a^m/n =
A
(^n√a)^m
23
Q
a^b+c =
A
a^b*a^c
24
Q
(a^b)^c
A
a^b*c
25
e is about
2.71828
26
standard form
f(x)= a(x-h)^2+k
27
h=
-b/2a
28
k=
plug h in function
29
if a>0, there is a
minimum value of k at x=h
30
if a<0, there is a
maximum value of k at x=h
31
graphing polynomial function steps
1) intercepts
2) sign check
3) end behavior
4) graph and label
32
when there is a multiplicity of 1, it
crosses the x axis
33
a^2+b^2=
(a+bi) (a-bi)
34
i^2 is
-1
35
use __ to determine real zeros of polynomials
long division
36
if c is a zero of p(x),
then x-c is a factor of p(x)
37
if a+bi is a zero of p(x),
a-bi is also a zero of p(x)
38
coefficient
the number ex: 4
39
term
the whole thing ex: 4x^2
40
domain constraints
1. 1/x ;x≠0
2. √x ;x≥0 *n is even
3. 1/√x ;x>0 *n is even *root is alone
4. logaX; x>0
41
graphing rational functions steps
1. factor
2. intercepts
- x int on numerator
- plug in 0 for y int
3. VA (x=)
- solve denominator
* sign chart
4. HA (y=)
5. graph and label
42
logarithmic properties
```
loga1= 0
logaA= 1
logaA^x=x
a^logaX= x
logaA^c= ClogaA
loga(AB) = logA+logB
loga(A/B)= logA-logB
```
43
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)
44
exponential and log equations steps
1) put exponential on one side
| 2) solve
45
trig functions
sin^2(x) + cos^2(x)= 1
tan^2(x)+1=sec^2(x)
1+cot^2(x)=csc^2(x)
46
degrees to radians
multiply π/180
47
radians to degrees
multiply 180/π
48
coterminal angles
add or subtract 2π (360 degrees) to given angle
49
trig functions of angles
SOH CAH TOA
| a^2+b^2=c^2
50
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: (-∞,∞)
51
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)
52
even-odd identities
```
sin(-x)= -sinx
cos(-x)= cosx
tan(-x)= -tanx
```
53
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
54
formula for sine
sin(x+y)= sinXcosY + cosXsinY
| sin (x-y)= sinXcosY - cosXsinY
55
formula for cosine
```
cos(x+y)= cosXcosY - sinXsinY
cos(x-y)= cosXcosY + sinXsinY
```
56
double-angle formulas
sine: sin2x= 2sinxcosx
cosine: cos2x= cos^2x-sin^2x
= 1-2sin^2x
= 2cos^2x-1
tangent: tan2x=sin2x/cos2x
57
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
```
58
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
```
59
tangent graph
```
atank(x-b)+c
period: π/k
d=period/4
*b in the middle
*2 asymptotes at the very ends
```
60
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
61
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
62
basic trig equations CHECK
1) factor
2) identities
3) formulas
- addition/subtraction
- double angle
* u substitution
63
if inverse function is on the outside, look at the
domain
64
if inverse function is on the inside, look at the
range
65
solving exponential/log equations: exponential
1) isolate exp
2) take loga
3) solve for variable
* no check
* sometimes we can factor
66
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
```
67
whenever u see NONLINEAR inequalities you must solve by ___
sign chart
68
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
69
basic graph of e
above x-axis
increasing from left to right
crosses (0,1)
70
basic graph of log
```
increasing from - y values to + y values
passes thru (1,0)
VA x=0
```
71
a^2 x b^2 =
(ab)^2
72
bad point
a point on the denominator of a non-linear inequality
73
zero point
real zeros *set equal to zero
74
how do u write factors that are 1 ± 2i
(x-(1-2i)) (x-(1+2i))
