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Concave up
Looks like a smile; slope increasing
Concave down
Looks like a frown, slope decreasing
Point of inflection
Where concavity changes
Even functions
Function is unchanged when reflected over y-axis
Odd functions
Function is unchanged when rotated 180 degrees around origin
End Behavior
Odd + Positive: limx→+∞ = +∞, limx→-∞ = -∞
Odd + Negative: limx→+∞ = -∞, limx→-∞ = +∞
Even + Positive: limx→+∞ = +∞, limx→-∞ = +∞
Even + Negative: limx→+∞ = -∞, limx→-∞ = -∞
End Behavior of Rational Functions
limx→0+ = +∞
limx→+∞ = 0
limx→-∞ = 0
limx→0- = -∞
Arc length formula
s=rθ
r is radians
θ is central angle in radians
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
Vertical asymptotes
y = tanx
x = π/2 + πn
y = cotx
x = πn
y = secx
same as tanx
y = cscx
same as o
Inverse Trigonometry
arcsinx: -π/2 ≤ x ≤ π/2
arccosx: 0 ≤ x ≤ π
arctanx: -π/2 ≤ x ≤ π/2
Converting from polar coordinates
x = rcosθ
y = rsinθ
Converting to polar coordinates
r = √x2 + y2
θ = tan-1 (y/x)
Semi-log plot
Vertical axis is logarithmically scaled, if graph looks linear, it is exponential
Residual plot
Pattern means inappropriate and no pattern means appropriate
Law of Sines
(a/sinA) = (b/sinB) = (c/sinC)
Angles (capital) are opposite to their sides (lowercase)
Law of Cosines
a2 = b2 + c2 - 2bc cosA
b2 = a2 + c2 - 2ac cosB
c2 = a2 + b2 - 2ab cosC
How to find domain
Do not divide by zero (exclude values that lead to this error)
Do not take the even root of negative numbers (numbers under the radical have to be positive, exclude numbers that make it negative)
Local extrema theorem
Polynomial function of degree n has at most n - 1 relative maxima/minima
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
Linearizing Exponential Data
Goes from y = aekx
Becomes lny = kx + lna
Follows y = mx + b form
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
Polar form of complex number
r(cosθ + isinθ)
Shorthand is rcisθ
Circle on polar graph
r = asinθ
r = acosθ
Cardoid equation
r = a(1 ± sinθ)
r = a(1 ± cosθ)
Limaçon equation
r = a ± bsinθ
r = a ± bcosθ
(a, b > 0, a =/ b)
a >/= 2b
r = a - bsinθ
Line equation
θ = K, passes through pole, slope = K
Spiral equation
r = aθ
Rose
r = asinnθ
r = acosnθ
n >/= 2
There are n petals when n is odd
There are 2n petals when n is even