AP Precalc

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29 Terms

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Concave up

Looks like a smile; slope increasing

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Concave down

Looks like a frown, slope decreasing

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Point of inflection

Where concavity changes

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Even functions

Function is unchanged when reflected over y-axis

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Odd functions

Function is unchanged when rotated 180 degrees around origin

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End Behavior

Odd + Positive: limx→+∞ = +∞, limx→-∞ = -∞

Odd + Negative: limx→+∞ = -∞, limx→-∞ = +∞

Even + Positive: limx→+∞ = +∞, limx→-∞ = +∞

Even + Negative: limx→+∞ = -∞, limx→-∞ = -∞

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End Behavior of Rational Functions

limx→0+ = +∞

limx→+∞ = 0

limx→-∞ = 0

limx→0- = -∞

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Arc length formula

s=rθ

r is radians

θ is central angle in radians

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Graph of trigonometric function

y = Asin( B (x + c)) + D

A is amplitude

The period is equal to 2π/B

The phase shift is C

The midline is y = D

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Vertical asymptotes

y = tanx

  • x = π/2 + πn

y = cotx

  • x = πn

y = secx

  • same as tanx

y = cscx

  • same as o

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Inverse Trigonometry

arcsinx: -π/2 ≤ x ≤ π/2

arccosx: 0 ≤ x ≤ π

arctanx: -π/2 ≤ x ≤ π/2

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Converting from polar coordinates

x = rcosθ

y = rsinθ

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Converting to polar coordinates

r = √x2 + y2

θ = tan-1 (y/x)

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Semi-log plot

Vertical axis is logarithmically scaled, if graph looks linear, it is exponential

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Residual plot

Pattern means inappropriate and no pattern means appropriate

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Law of Sines

(a/sinA) = (b/sinB) = (c/sinC)

Angles (capital) are opposite to their sides (lowercase)

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Law of Cosines

a2 = b2 + c2 - 2bc cosA

b2 = a2 + c2 - 2ac cosB

c2 = a2 + b2 - 2ab cosC

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How to find domain

  1. Do not divide by zero (exclude values that lead to this error)

    1. Do not take the even root of negative numbers (numbers under the radical have to be positive, exclude numbers that make it negative)

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Local extrema theorem

Polynomial function of degree n has at most n - 1 relative maxima/minima

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Point of Inflection Theorem

Polynomial function of degree n, where n greater than or equal to 2, has at most n - 2 points of inflection, graph of odd degree has at least 1 point of inflection

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Linearizing Exponential Data

Goes from y = aekx

Becomes lny = kx + lna

Follows y = mx + b form

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Rectangular-Polar Conversion

arctan(y/x) for x > 0, y > 0

π - arctan(|y/x|) for x < 0, y > 0

arctan(|y/x|) + π for x < 0, y < 0

2π - arctan(|y/x|) for x > 0, y < 0

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Polar form of complex number

r(cosθ + isinθ)

Shorthand is rcisθ

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Circle on polar graph

r = asinθ

r = acosθ

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Cardoid equation

r = a(1 ± sinθ)

r = a(1 ± cosθ)

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Limaçon equation

r = a ± bsinθ

r = a ± bcosθ

(a, b > 0, a =/ b)

a >/= 2b

r = a - bsinθ

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Line equation

θ = K, passes through pole, slope = K

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Spiral equation

r = aθ

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Rose

r = asinnθ

r = acosnθ

n >/= 2

There are n petals when n is odd

There are 2n petals when n is even