AP Pre Calc Flash Cards

The a in g(x)= a·f (b(x±h))±k

Vertical dilation by a factor of |a|

Reflection over x-axis if a<0

The h in g(x) = a·f (b(x±h))±k

Horizontal translation

Left when x + h, right when x - h

The k in g(x) = a·f (b(x±h))±k

Vertical translation

Up when k>0, down when k<0

The b in g(x) = a·f (b(x±h))±k

Horizontal dilation by a factor of |1/b|

Reflection over y-axis if b<0

Average rate of change between (a,f(a)) and (b,f(b))

f(b)-f(a)/b-a

Where is a function positive?

When the y-coordinates are above the x-axis.

Where is a function negative?

When the y-coordinates are below the x-axis.

What is a point of inflection?

The ordered pair where concavity changes.

What justifies an increasing rate of change?

When a function is concave up.

What justifies a decreasing rate of change?

When a function is concave down.

What does it mean if c is odd in f(x) = a(x-b)^c

C is a zero with odd multiplicity

The graph of f will cross the x-axis at x = b

What does it mean if c is even in f(x) = a(x-b)^c

C is a zero with even multiplicity

The graph of f will touch the x-axis and turn at x = b

What is an even function?

When f(-x) = f(x)

Appears symmetric about the y-axis

What is an odd function?

When f(-x) = f(x)

Appears symmetric about the y-axis

Notation for end behavior as inputs decrease without bound

lim f(x) negative infinity

Notation for end behavior as inputs increase without bound

lim f(x) positive infinity

Horizontal asymptote test when r(x) =xⁿ degree polynomial / zⁿ degree polynomial

If n < m, H.A at y=0

If n = m, H.A at y = L.C num./L.C Denom.

If n > m, no H.A

When does a rational function have a slant asymptote?

When the degree of the poly in the numerator is exactly one more than the degree of the poly in the denominator

Where does a rational function have a hole? hint (x-a)/(x-a)(x-b)

When a factor cancels in numerator and denominator (unless if covered by a V. A.)

Where does a rational function have a vertical asymptote?

When a factor is a zero of the denominator after canceling

Standard form of an arithmetic sequence?

aₙ=a₁+(n-1)d

Standard form of a geometric sequence

gₙ=g₁×^(n-k)

Exponential decay in y=a×bⁿ

When 0 < |b| < 1

As the inputs increase, the outputs are moving toward the x-axis

Exponential growth in y=a×bⁿ

When |b| > 1

As the inputs increase, the outputs are moving away from the x-axis

logₙx+logₙy=

logₙ(xy)

nlogₙx=

logₙxⁿ

logₙx-logₙy=

logₙ(x/y)

logₙx/logₙy

log_y of x

Pythagorean Trig Identity

sin²x+cos²x=1

sin (α+β)=

sinαcosβ+cosαsinβ

sin(α-β)=

sinαcosβ-cosαsinβ

cos(α+β)

cosαcosβ-sinαsinβ

cos(α-β)

cosαcosβ+sinαsinβ

Vertical asymptotes of f(x) = a(secbx)+d

bx=π/2 + πk

Vertical asymptotes of f(x) = a(cscbx)+d

bx=πk

Vertical asymptotes of f(x) =a(cotbx)+d

bx=πk

Vertical asymptotes of f(x)=a(tanbx)+d

bx=π/2 + πk

Period of y =a×sin(b(x±c))±d or y=a×cos(b(x±c))±d

2π/b

Period of y = a×tan(b(x±c))±d or y = a×cot(b(x±c))±d

π/b

How to convert (r,θ)→(x,y)

x = rcosθ

x = rsinθ

How to convert (x,y) →(r,θ)

x²+y²=r²

tanθ=y/x, if x>0

tanθ=y/x+π if x<0

How to convert a+bi to (rcosθ)+i(rsinθ)

a²+b²=r²

tanθ= b/a if a>0

tanθ= b/a+π if a<0

Arc length

θ×r

Range of y=sin⁻¹x

[-π/2, π/2]

Range of y=cos⁻¹x

[0,π]

Range of y=tan⁻¹(x)

(-π/2,π/2)

In a polar function, when is the distance from the origin increasing?

When r=f(θ) is positive and increasing or r=f(θ) is negative and decreasing

in a polar function, when is the distance from the origin decreasing?

When r=f(θ) is positive and decreasing or when r=f(θ) is negative and increasing

What is error?

|predicted value (from regression) - actual value| or the opposite sign of the residual

Compare period vs. frequency in a trig function

Priod is the length required for one full cycle of outputs

Frequency is the reciprocal or period

Frequency is how many cycles per unit of time

What type of function has constant first differences over equal-length inputs?

A linear function

What type of function has constant rate of changes in first differences over equal-length inputs?

A quadratic function

What type of function has constant third differences over equal-length inputs?

A cubic function

What type of function has proportional outputs over equal-length inputs?

An exponential function

What type of function has proportional inputs over equal-length outputs?

A logarithmic function