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1

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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4

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

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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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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 valuea 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)

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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 | g(x) = cos | h(x) = tan | |
---|---|---|---|

| 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 |

| standard position angle | standard position angle | standard position angle |

| x→[-1,1] | x→[-1,1] | x→[-∞,∞] |

| 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 + b** i** = r(cosθ +

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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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