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RECOGNIZING CHANGE PATTERNS IN FUNCTIONS (starts on next card)

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RECOGNIZING CHANGE PATTERNS IN FUNCTIONS (starts on next card)

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linear

rate of change is constant on any interval

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quadratic

2nd differences of output values are constant over consecutive equal-length input value intervals

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polynomial (degree n)

nth differences of output values are constant over consecutive equal-length input value intervals

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exponential

output values are proportional over equal-length input value intervals

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logarithmic

proportional input values result in constant change in output values

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RATES OF CHANGE (starts on next card)

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average rate of change (AROC)

the aroc of f(x) over the interval [a, b] is the slope of the line between the points

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rate of change (ROC)

roc of f at a point refers to the slope of the graph at that point

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BEHAVIOR OF RATIONAL FUNCTIONS (starts on next card)

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SEQUENCE FORMULAS (starts on next card)

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arithmetic

<p></p>
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geometric

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PROPERTIES OF EXPONENTS (flip this card)

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PROPERTIES OF LOGARITHMS (flip this card)

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TRANSFORMATIONS (starts on next card)

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g(x) = a f(b(x+h))+k

(vertical changes)

stretch if |a|>1

compression if |a|<1

reflection if a is negative

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g(x) = a f(b(x+h))+k

(horizontal changes)

stretch if |b|>1

compression if |b|<1

reflection if b is negative

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<p>FUNCTION BEHAVIORS (flip this card)</p>

FUNCTION BEHAVIORS (flip this card)

-∞ to D: f(x) is concave down. roc is decreasing.

D to ∞: f(x) is concave up. roc is increasing.

C to F: f(x) is decreasing. roc is negative.

F to ∞: f(x) is increasing. roc is positive.

A to E: f(x) is positive

E to G: f(x) is negative

C and F: extrema (C - relative max; F - relative min)

D: point of inflection (change in concavity)

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INVERSE FUNCTIONS (starts on next card)

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numerically

if f(a) = b then f-1(b)=a

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graphically

f-1(x) is the reflection of f(x) over the y=x

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algebraically

to find f-1(x), switch x and y on f(x) and solve for x

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verbally

f-1(x) uses the opposite operations of f(x) in the reverse order

  • f-1(x) is a function if every output value of f(x) comes from a unique input value

  • a function that is either strictly increasing or strictly decreasing is invertible

  • the domain of f-1(x) is the range of f(x) and vice versa

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extra note?? or smth ig??? idk atp

f-1(f(x)) = f(f-1(x)) = x

for all x in the domain of either function

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IDENTITIES (starts on next card)

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pythagorean

  • sin²θ + cos²θ = 1

  • tan²θ + 1 = sec²θ

  • cot²θ + 1 = csc²θ

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reciprocals (pythagorean)

<p></p>
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quotients (pythagorean)

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angle sum/difference

  • sin (a+b) = (sin a)(cos b) + (cos a)(sin b)

  • sin (a-b) = (sin a)(cos b) - (cos a)(sin b)

  • cos (a+b) = (cos a)(cos b) - (sin a)(sin b)

  • cos (a-b) = (cos a)(cos b) + (sin a)(sin b)

SINE COSINE COSINE SINE COSINE COSINE SINE SINE AHHAHAHHHHHHHHH ok sorry

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double angle

  • sin 2θ = 2 sinθ cosθ

  • cos 2θ = cos²θ - sin²θ

    • = 2 cos²θ - 1

    • = 1 - 2 sin²θ

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SINE/COSINE GRAPH PROPERTIES (starts on next card)

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f(x) = a sin (b(x+c)) + d

  • y = sinx: starts at midline and does upward

  • y = cosx: starts at maximum and goes downward

|a| = amplitude d = midline

|a| + d = maximum -e = phase shift

|a| - d = minimum (2π)/b = period

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SPECIAL RIGHT TRIANGLES (flip this card)

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INVERSE TRIG FUNCTIONS (can be written as sin-1(x) or arcsin(x) (flip this card)

f(x) = sin-1x

g(x) = cos-1x

h(x) = tan-1x

input

ratio of sides or y-coordinate on unit circle

ratio of sides or x-coordinate on unit circle

ratio of sides or slope of terminal side of θ

output

standard position angle

standard position angle

standard position angle

domain

x→[-1,1]

x→[-1,1]

x→[-∞,∞]

range

y→[(-π/2), (π/2)] (quad 1 and 4)

y→[0, π] (quad 2 and 3)

y→[(-π/2), (π/2)] (quad 1 and 4)

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POLAR COORDINATES (starts on next card)

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rectangular (x,y) → polar (r,θ)

  • r = √x² + y²

  • tan-1(|y/x|) = reference angle (use quadrant and reference angle to find θ)

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polar (r,θ) → rectangular (x,y)

  • x = r cosθ

  • y = r sinθ

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polar form of a complex number

a + bi = r(cosθ + isinθ)

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PASCAL’S TRIANGLE (flip this card)

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JUSTIFICATIONS/FRQ TERMINOLOGY (starts on next card)

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a function is increasing over an interval of its domain if:

for all a and b in the interval, if a < b, then f(a) < f(b)

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a function is decreasing over an interval of its domain if:

for all a and b in the interval, if a < b, then f(a) > f(b)

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YOU GOT THIS WOOOOOO!!!!!!

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