exam Flashcards
(82 cards)
1
Q
One-to-one function
A
each x value pairs to exactly one UNIQUE y value
- Passes HLT & VLT
(No repeated x or y values in a table)
2
Q
Function
A
each x value pairs to exactly one y value
- Passes VLT
(No repeated x values in a table)
3
Q
Degree of a polynomial
A
the greatest degree of any term in the polynomial
4
Q
Leading term
A
the term with the highest degree or exponent
5
Q
How many turning points (min or max values) does a polynomial have?
A
n-1
n = the degree of the polynomial
6
Q
Inflection point
A
= Leading exponent - 2
Point where concavity changes
(CU to CD or CD or CU)
7
Q
Asymptote
A
a line that a graph approaches but never crosses
8
Q
What kind of asymptote(s) does an exponential function have?
A
horizontal asymptote
9
Q
What kind of asymptote(s) does a logarithmic function have?
A
vertical asymptote
10
Q
Vertical stretch
A
the whole function, f(x), is multiplied by a (where a > 1)
11
Q
Vertical compression
A
the whole function, f(x) is multiplied by a (where a is a fraction: 0 < a < 1)
12
Q
Horizontal stretch
A
- x is multiplied by a (where a is a fraction: 0 < a < 1)
- graph stretches away from the y-axis
13
Q
Horizontal compression
A
- x is multiplied by a (where a > 1)
- graph compresses toward the y-axis
14
Q
how to find the inverse of a function
A
- Change f(x) to y
- Swap x & y
- Isolate y
- y becomes f^-1(x)
15
Q
A function is even if
A
f(-x) = f(x)
- It is symmetric over the y-axis
16
Q
A function is odd if
A
f(-x) = -f(x)
- It is symmetric rotationally over the origin
17
Q
Complex zeroes
A
When solving for all the zeroes, there will be a negative under the square root. Replace it with i
- Its conjugate is ALSO a zero
18
Q
Complex conjugate
A
Complex numbers: a+bi and a-bi
- If a function has a zero at a+bi, it ALSO has a zero at a-bi
19
Q
Multiplicity of a zero
A
Number of times a zero’s factor occurs in a polynomial
- If odd, the line passes through that zero
- If even, the line will be tangent to the x-axis (bounce off) at that zero
20
Q
End behavior (EB)
A
The behavior as x approaches positive or negative infinity:
- EB of an even function is the same for its -∞ & ∞
- EB of an odd function is the opposite for its -∞ & ∞
21
Q
Rational function
A
22
Q
Linear, quadratic, or exponential from table of values?
A
- Linear = First difference in y is constant
- Quadratic = Difference in y is not constant, but the second difference (difference between successive first differences) is constant
- a = second difference / 2
- Exponential = Difference in y follows a similar pattern to the y values
23
Q
End behavior of a rational function
A
- Numerator degree > denominator degree:
- EB matches the EB of the quotient of the leading terms - Numerator degree = denominator degree:
- EB approaches the horizontal asymptote (= ratio of leading terms) in both directions - Numerator degree < denominator degree:
- EB approaches the horizontal asymptote y = 0 in both directions
24
Q
e
A
2.718
25
Continuous growth/decay formula
f(t) = a * e^Kt
1. a is the initial amount
2. e is the constant natural base (the letter e, like pi)
3. k the rate of change per unit of time (DON'T add 1 if, e.g., 5%)
4. t is the units of time
26
Log and exponential functions are _____ of each other
inverse functions
27
y = log(base b)x is the inverse of...
y = b^x
28
if log(base b)C = a, then
b^a = C
29
log (x*y) =
log (x) + log (y)
30
log (x/y) =
log x - log y
31
log (x^P) =
P * log (x)
32
log (3^x) =
x * log (3)
33
log(base e) =
ln
34
if log(base a)x = log(base a)y, then
x=y
35
log(base a)1 =
0, because a^0 = 1
36
log(base a)a =
1, because a^1 = a
37
log(base b) of a number ≤ 0
does not exist (gives a calculation error)
38
How to find the domain/range of the inverse of a function
The range of the inverse function is the domain of the original
The domain of the inverse is the range of the original
39
When you get solutions to log functions, you need to...
plug them in because they may be extraneous
40
The 6 things to know about log graphs
1. Domain is positive real #s
2. x-int is (1, 0)
3. y-int DOESN'T EXIST, as log(0) doesn't exist (x = 0 is an asymptote)
4. If base > 1, it's an increasing function
5. If 0 < base < 1, it's a decreasing function
6. Always either increasing or decreasing, meaning no extrema or relative extrema and no points of inflection
41
Extrema and relative extrema
Extrema: Absolute maxes/mins
Relative extrema: Local maxes/mins
42
How to solve a logarithmic equation with a constant (steps)
1. Bring logs with the same base to the same side, and combine into 1 log
2. Get into the form: log(base b)c = a
3. Rewrite in exponential form: b^a = c and solve
4. Check the solution(s) in the original equation (extraneous if it creates a log of zero or a negative number)
43
Continuous exponential change formula
f(x) = A(e)^rt
A = initial amount
r = rate of change per unit of time (e.g., if +1%, it's 1.01)
t = time
44
When you have to make an exponential function for something in nature (e.g., bacterial growth)...
USE BASE e
45
Residual
observed value - predicted value
46
Residual plot
x-axis = independent variable
y-axis = residual values
- If points are randomly dispersed, a linear model will fit best
- If points have a pattern, a non-linear model will fit best
47
Term of arithmetic sequence formula
An = Ak + d(n - k)
An = the nth term (you are trying to find)
Ak = the kth term in the sequence
n = the desired term number
k = the given term number (often 1)
d = the common difference
48
Term of Geometric sequence formula
An = Ak * r^(n-k)
An = the nth term (you are trying to find)
Ak = the kth term in the sequence
n = the desired term number
k = the given term number (often 1)
r = the common ratio
49
In the xy-plane, sin θ = ___
sin θ = vertical displacement/r,
where r is the radius
50
In the xy-plane, cos θ = ___
cos θ = horizontal displacement/r,
where r is the radius
51
On the unit circle, tan θ = ___
tan θ = m = y/x = sin θ/cos θ
where the terminal side and the unit circle intersect at (x, y)
because the hypotenuse (radius) = 1
52
Formula for finding point P using the radius and angle
(r cosθ, r sinθ)
53
Types of Discontinuity
1. Infinite/asymptotic
2. Removable/point
3. Jump
54
AP Exam: Round to ___ decimal places
3
THE THIRD ONE!
55
When do you need to plug in your solutions to check for extraneous solutions
When solving:
1. Radical equations (when you get a solution after squaring both sides to get rid of a square root)
2. Logarithmic equations (with logs)
56
graphing tan(x)
1. graph the vertical asymptotes (at pi/2 and 3pi/2)
and their coterminals
2. graph the 3 points between the asymptotes
3. has a period of pi
57
cos(x) is a(n) _____ function, unless...
even, unless it is transformed (then it might or might not be even)
Meaning f(x) = f(-x), thus cos(x) = cos(-x),
e.g., cos(π/4) = √2/2 and cos(-π/4) = √2/2
58
sin(x) is a(n) _____ function, unless...
odd, unless it is transformed (then it might or might not be odd)
Meaning -f(x) = f(-x), thus -sin(x) = sin(-x),
e.g., -sin(π/4) = -√2/2 and sin(-π/4) = -√2/2
59
Domain and range of tan(x)
D: {x | x ≠ n * π/2}, where n = any integer
R: All real numbers
60
Tan function is an_________ function unless...
Odd function unless it is transformed (then it may are may not be odd)
61
Sinusoidal function equation and information
y = a sin(b(x - c/b)) + d,
|a| is the amplitude (if negative, it's reflected over the y-axis)
b is the horizontal stretch/compression
c/b is the phase shift (horizontal translation)
d is the midline (vertical translation)
62
cosecant
Reciprocal of sin (meaning 1/sin)
E.g., sin A = a/h, csc A = h/a, where h = hypotenuse
63
Steps to graph a cosecant function
1. Graph the sin function with the same modifications
2. Draw asymptotes where it crosses the midline
3. Draw 'U's and '∩'s at maxima and minima, respectively
64
Secant (sec)
Reciprocal of cos (meaning 1/cos)
E.g., cos B = b/h, sec B = h/b, where h = hypotenuse
65
Steps to graph a secant function
1. Graph the cos function with the same modifications
2. Draw asymptotes where it crosses the midline
3. Draw 'U's and '∩'s at maxima and minima, respectively
66
Cotangent
Reciprocal of tan (1/tan)
67
Steps to graph a cotangent function
1. Create a table of values for tan x
2. Use the resulting output values to calculate corresponding 1/tan values
3. Graph those
68
Polar coordinates
Points have coordinates (r,θ),
- | r | is the radius
- | θ | is the counterclockwise angle
- If r is negative, the point is reflected over the pole (origin)
- If θ is negative, it is the clockwise angle
69
Complex plane
Graphing numbers in the form of a + bi
Horizontal axis: a
Vertical axis: b
Points become (a, b)
70
Complex number to polar form
(a, b) = (rcosθ, rsinθ), making Z = a + bi = r(cosθ + isinθ)
where r = radius (it's also factored out of the sum)
71
Graphs of polar functions
r = f(θ)
- θ is the input, r is the output
72
Petal counts for graphs of polar functions
If b is even: there will be 2b petals
If b is odd: there will be b petals
Where the function takes the form:
r(θ) = a*trig_function(bx)
73
Limacon
a/b < 1,
where...
- r(θ) = a ± b*sin(x)
- or r(θ) = a ± b*cos(x)
- Visually: Limacon kind of looks like a lemon (oval/circle with inner loop)
74
Cardioid
a/b = 1,
where...
- r(θ) = a ± b*sin(x)
- or r(θ) = a ± b*cos(x)
- Visually: Cardioid is a circle with a punch into the bottom that touches the pole (origin)
75
Dimpled cardioid
1 < a/b < 2,
where...
- r(θ) = a ± b*sin(x)
- or r(θ) = a ± b*cos(x)
- Visually: Dimpled cardioid is a circle with a punch into the bottom that Doesn't quite touch the pole (origin)
76
Why does the AROC produce an over-/under-estimate?
If f(x) is concave up over the interval, the secant line will be above f(x), producing an overestimate
If f(x) is concave down over the interval, the secant line will be below f(x), producing an underestimate
77
A function is invertible if and only if
It has no repeated output values. In other words, every output value is mapped from only one unique input value.
GIVE EXAMPLES IN FRQ
78
Pythagorean identities
sin^2x+cos^2x=1
tan^2x+1=sec^2x
1+cot^2=csc^2
79
tan and cot identities
tanx = sinx/cosx
cotx = cosx/sinx
80
Periods of trigonometric functions
sin and cos (and their reciprocals) = 2pi
tan and cot = pi
81
double angle identities
sin2x = 2sinxcosx
cos2x = cos^2x-sin^2x OR 1-2sin^2 OR 2cos^2x-1
tan2x = 2+tanx/1-tan^2x
82
sum and difference identities
sin(x+y) = sinxcosx + cosxsiny
sin(x-y) = sinxcosx - cosxsiny
cos(x+y) = coxcosy - sinxsiny
cos(x-y) = cosxcosy + sinxsiny
tan(x+y) = tanx +tany/1-tanxtany
tan(x-y) = tanx-tany/1+tany
