AP Pre-Calculus Flashcards

The average rates of change of a linear function are

Constant

The average rates of change of a quadratic function

Are changing at a constant rate OR follow a linear pattern

A function is quadratic if over equal-length input intervals, output values

Change by a constant second difference

A function is logarithmic if as input values ____, output values change ____

multiplicity; additively

A function is exponential if as input values change ____, output values change ____

additively; multiplicatively

A function is linear if over equal-length input intervals, output values

Change by a constant amount

A function f is concave down if

the rates of change of f are decreasing

A positive rate of change indicates that the function output is

Increasing

The slope of a function at any given point gives

the rate of change of the function at that input

A negative rate of change indicates that the function output is

decreasing

Average rate of change of f on the interval [a,b]

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

f(-x) = -f(x)

Odd Function

Multiplicity

The number of times a factor occurs in a polynomial

f(-x) = f(x)

Even Function

One-to-one Function

Function where each input has a unique output

If x = a is a real zero of a polynomial with an odd multiplicity, then

The graph of the polynomial passes through the x axis at x = a

If x = a is a real zero of a polynomial with an even multiplicity, then

the graph of the polynomial touches the x axis at x = a but does not cross it.

A polynomial of degree n has

Exactly n complex zeros (real or imaginary), constant nth difference, at most n - 1 extrema

Point of Inflection

a point on the graph where the curvature changes from concave up to concave down or vice versa.

A function f(x) = abˣ demonstrates exponential decay if

0 < b < 1

Key features of y = bˣ where b>1

Domain: all real numbers

Range: y > 0

Horizontal Asymptote at y = 0

Increasing and concave up over entire domain

Key features of y = logb(x) where b>1

Domain: x > 0

Range: All real numbers

Vertical Asymptote at x = 0

Increasing and concave down over entire domain

A function f is decreasing on an interval if

As the input values increase, the output values always decrease OR for all a and b in the interval, if a<b, then f(a) > f(b)

A function f is concave up if

the rate of change of f are increasing

A function f(x) = abˣ demonstrates exponential growth if

b > 1

End behavior of a polynomial f with odd degree and negative leading coefficient

lim f(x) = -inf when x approaches inf

lim f(x) = inf when x approaches -inf

End behavior of a polynomial f with an even degree and a negative leading coefficient

lim f(x) = -inf when x approaches inf

lim f(x) = -inf when x approaches -inf

End behavior of a polynomial f with an even degree and a positive leading coefficient

lim f(x) = inf when x approaches inf
lim f(x) = inf when x approaches -inf

End behavior of a polynomial f with an odd degree and a positive leading coefficient

lim f(x) = inf when x approaches inf
lim f(x) = -inf when x approaches -inf

If a ration function, f, has a horizontal asymptote at y = b, then

The ratio of leading terms is a constant, b, and lim f(x) = b when x approaches both inf and -inf

A rational function has a zero at x = a if

x = a is a zero of the numerator but NOT the denominator

A rational function has a hole at x = a if

x = a is a zero of both the numerator and the denominator

A rational function has a vertical asymptote at x = a if

x = a is a zero of the denominator but NOT the numerator.

For rational functions, a slant asymptote occurs when

the degree of the numerator is one higher than that of the denominator.

A relative maximum occurs when a function f

Changes from increasing to decreasing

Absolute Minimum

The least output of a function

A relative minimum occurs when a function f

changes from decreasing to increasing.

Absolute Maximum

The greatest output of a function

To determine the end behavior of a rational function

Analyze the ratio of leading terms

A function f is increasing on an interval if

As the input values increase, the output values always increase OR for all the a and b in the interval, if a>b, then f(a)<f(b)

Key features of y = sinx

Domain: all reals

Range: [-1,1]

Period: 2π

Amplitude: 1

Midline: y = 0

Passes through (0,0)

Key Feature of y = cosx

Domain: all reals
Range: [-1,1]
Period: 2π
Amplitude: 1
Midline: y = 0
Passes through (0,1)

Domain and range of y = arctanx

Domain: (-inf, inf)

Range: (-π/2, π/2)

Domain and range of y = arccosx

Domain: [-1, 1]

Range: [0, π]

Domain and range of y = arcsinx

Domain: [-1, 1]

Range: [-π/2, π/2]

y = tan(bx) has an period of

π/b

f(x) = cotx has a vertical asymptotes at

x = kπ, where k is an integer

f(x) = tanx has vertical asymptotes at

x = (π/2) + kπ, where k is an integer

tanx gives the ____ of the terminal ray of x

slope

Determine the amplitude, period, midline, and phase shift of f(x) = asin(b(x-c)) + d

Amplitude: a

Period: 2π/b

Midline: y = d

Phase Shift: c units to the right

Horizontal dilation by a factor of 1/c

f(cx)

Reflection over the y axis

f(-x)

Horizontal translation c units to the left

f(x + c)

Horizontal translation c units to the right

f(x - c)

Vertical translation up c units

f(x) + c

Vertical translation down c units

f(x) - c

Vertical dialation by a factor of c

cf(x)

Reflection over the x axis

-f(x)

cos(2x)

cos²(x) - sin²(x)

2cos²(x) - 1

1 - 2sin²(x)

logₐ(1)

0

eᵃᴸⁿᵇ

bᵃ

bˣ⁺ᶜ

bˣ(bᶜ)

bˣ⁻ᶜ

bˣ/bᶜ

logᵦ(m/n)

logᵦ(m) - logᵦ(n)

logᵦ(b)

1

logᵦ(mn)

logᵦ(m) + logᵦ(n)

logᵦmᵏ

klogᵦ(m)

cos (a+b)

cos(a)cos(b) - sin(a)sin(b)

cos(a-b)

cos(a)cos(b) + sin(a)sin(b)

sin(a+b)

sin(a)cos(b) + sin(b)cos(a)

sin(a-b)

sin(a)cos(b) - sin(b)cos(a)

sin(2x)

2sin(x)cos(x)

sin²x + cos²x

1

1 + cot²x

sec²x

tan²x + 1

sec²x

tan(2x)

2tan(x)/(1-tan²(x))

tan(u-v)

tan(u) - tan(v) / 1 + tan(u)tan(v)

tan(u+v)

tan(u) + tan(v) / 1 - tan(u)tan(v)

rcos(θ)

x

rsin(θ)

y

A polar function r=f(θ) is decreasing if

As θ increases, r decreases

The distance between a point on a polar function r=f(θ) and the origin is decreasing if

r is positive and decreasing or r is negative and increasing

The distance between a point on a polar function r=f(θ) and the origin is increasing if

r is positive and increasing or r is negative and decreasing

A polar function r=f(θ) is increasing if

As θ increases, r increases

tan(θ)

y/x

r²

x² + y²

A model is considered appropriate for a data set if the residual plot

shows no obvious patterns

f and g are inverse functions if

f(g(x)) = x and g(f(x)) = x

A negative residual indicates that the predicted value is an

Overestimate

Explicit rule for the nth term of an arithmetic sequence, given common difference d and the aₖ term

aₙ = aₖ+d(n-k)

In a semi-log plot where the y axis is logarithmically scaled, exponential functions will appear

linear

Residual

Actual value - predicted value

Error (in a model)

Predicted value - actual value

A positive residual indicates that the predicted value is an

underestimate

Explicit rule for the nth term of a geometric sequence given common ratio r, and the aₖ term

aₙ = aₖr^(n-k)

If the y-axis is logarithmically scaled, then

Equal sized increments on the y-axis represent proportional changes in the output variable

what does the constant e represent

The base rate of growth for all continuous growth processes, approximately equal to 2.71828.