AP Precalculus Review Flashcards

Flashcards for reviewing key concepts in Polynomial, Rational, Exponential, Logarithmic, Trigonometric, and Polar Functions.

What is the formula for Average Rate of Change (AROC) on an interval [a, b]?

AROC = (f(b) - f(a)) / (b - a)

What does a positive rate of change indicate about a function?

Increasing function

What does a negative rate of change indicate about a function?

Decreasing function

What does concave up indicate about the rate of change?

Rate of change is increasing

What does concave down indicate about the rate of change?

Rate of change is decreasing

What characterizes a point of inflection?

Rate of change changes from increasing to decreasing or decreasing to increasing

What is the test for an odd function?

f(-x) = -f(x)

What is the test for an even function?

f(-x) = f(x)

What type of symmetry does an even function exhibit?

Reflects over the y-axis

What type of symmetry does an odd function exhibit?

Passes through the origin

What is the justification phrase related to function behavior?

Over equal-length input-value intervals…

What happens at a zero with multiplicity 1?

Crosses x-axis (linearly)

What happens at a zero with multiplicity 2?

Bounces (quadratically)

What happens at a zero with multiplicity 3?

Bends (cubically)

How do you find the x-intercept(s)?

Let y = 0

How do you find the y-intercept?

Let x = 0

What is a complex conjugate?

If a + bi is a factor, then a - bi is also a factor

Describe end behavior for odd degree polynomials.

End-behaviors are opposites

Describe end behavior for even degree polynomials.

End-behaviors are the same

What does a positive leading coefficient indicate about end behavior?

Up on the right

What does a negative leading coefficient indicate about end behavior?

Down on the right

In the transformation F(x) = a * f[b(x - h)] + k, what does 'a' control?

Vertical reflection and dilation

In the transformation F(x) = a * f[b(x - h)] + k, what does 'b' control?

Horizontal dilation

In the transformation F(x) = a * f[b(x - h)] + k, what does 'h' control?

Horizontal translation

In the transformation F(x) = a * f[b(x - h)] + k, what does 'k' control?

Vertical translation

How can you analyze table problems to identify a linear function?

If both input and output values change consistently

How can you analyze table problems to identify a quadratic function?

If input changes consistently and the 2nd Differences of output values are equal

How can you analyze table problems to identify a cubic function?

If input changes consistently and the 3rd Differences of output values are equal

How can you analyze table problems to identify an exponential function?

If input changes consistently and output values change proportionally

How can you analyze table problems to identify a logarithmic function?

If input values change proportionally and output values change consistently

How do vertical asymptotes occur in rational functions?

When a factor in the denominator does not cancel out with factors in the numerator

When does a horizontal asymptote occur at y=0?

If the degree of the numerator is less than the degree of the denominator (n < m)

When does a horizontal asymptote occur at y=a/b?

If the degree of the numerator equals the degree of the denominator (n = m)

When do you not have a Horizontal Asymptote?

If the degree of the numerator is greater than the degree of the denominator (n > m)

How do you find slant/oblique asymptotes?

Use long division (ignoring the remainder) to find the equation if the degree of the numerator is one degree larger than the degree of the denominator

How do holes occur in rational functions?

When a common factor cancels out after simplifying the rational expression

How do you find the location of a hole?

Evaluate the simplified expression at the x-value of the hole

Give the formula for arithmetic sequences.

𝑎𝑛 = 𝑎𝑘 + 𝑑(𝑛 − 𝑘) or 𝑎𝑛 = 𝑎0 + 𝑑𝑛

Give the formula for geometric sequences.

𝑔𝑛 = 𝑔0 (𝑟)𝑛 or 𝑔𝑛 = 𝑔𝑘 * (𝑟)^(𝑛−𝑘)

Give the algebraic formula for Linear functions.

f(x) = mx+b

Give the algebraic formula for Exponential functions.

f(x) = a * b^(x-h) + k

Describe the characteristics of Exponential Functions.

Concave up or concave down everywhere, horizontal asymptote at y = k. either increasing or decreasing everwhere.

Exponential function values for consistent input values change in what manner?

Proportionally

What is the exponential to logarithmic form conversion?

b^p = n converts to log_b(n) = p

Logarithmic function input values for consistent output values change in what manner?

Proportionally

What are the basic properties of Logarithms?

logb(1) = 0, logb(b) = 1, logb(b^y) = y, b^(logb(x)) = x

What are the basic properties of Natural Logarithms?

ln(1) = 0, ln(e) = 1, ln(e^y) = y, e^ln(x) = x

State the Logarithm Product Property.

logb(MN) = logb(M) + log_b(N) or ln(MN) = ln(M) + ln(N)

State the Logarithm Quotient Property.

logb(M/N) = logb(M) - log_b(N) or ln(M/N) = ln(M) - ln(N)

State the Logarithm Power Property.

logb(N^p) = p * logb(N) or ln(N^p) = p * ln(N)

How do inverse functions relate?

A function in which input values and output values are “switched,” such that an ordered pair for a function 𝑓 is (𝑎, 𝑏) becomes (𝑏, 𝑎) on the inverse function 𝑓−1(𝑥).

What is the result of composition of inverse functions?

Inverse functions "undo" each other, f(f^{-1}(x)) = x and f^{-1}(f(x)) = x

What happens to an exponential model on a Semi-Log Plot?

A data set that behaves in an exponential model will appear linear when the y-axis is logarithmically scaled.

If 𝒇(𝒙) is even, then:

𝒇(−𝒙) = 𝒇(𝒙)

If 𝒇(𝒙) is odd, then:

𝒇(−𝒙) = −𝒇(𝒙)

In what quadrants are ALL trig functions positive?

Quadrant I

In what quadrants are Sine and Cosecant positive?

Quadrant II

In what quadrants are Tangent and Cotangent positive?

Quadrant III

In what quadrants are Cosine and Secant positive?

Quadrant IV

What is the reciprocal identity of csc(x)?

csc(x) = 1/sin(x)

What is the Pythagorean identity relating sin and cos?

sin^2(x) + cos^2(x) = 1

What is the double angle formula for sin(2x)?

sin(2x) = 2sin(x)cos(x)

Relationship between rectangular coordinates (x,y) and polar coordinates?

(x, y) = (rcos(theta), rsin(theta))

How do you convert from rectangular to polar coordinates?

x^2 + y^2 = r^2, tan(theta) = y/x (x != 0)

In y = A*sin(B(x+C)) + D, what variable indicates Amplitude?

A

In y = A*sin(B(x+C)) + D, what variable indicates horizontal Phase Shift?

C

In y = A*sin(B(x+C)) + D, what variable indicates Vertical Shift?

D

State the formula for the period of a sinosoidal function.

Period = 2pi/B

What does an 'a' value less than zero in the sinosoidal transformations represent?

Reflects graph over x-axis

Tangent Function: know that the equations for Vertical Asymptotes are…

θ ≠ 𝜋/2 + kπ where k is any integer

Write the equations for circle polar equations with radius a

r = a cos θ or r = a sin θ

What equation/ratio indicates the shape of Cardiods?

r = a ± b cos θ --> a/b = 1

What equation/ratio indicates the shape of Convex Limacon (without an inner loop)?

r = a ± b cos θ --> a/b > 2

What equation/ratio indicates the shape of Limacon (with an inner loop)?

r = a ± b cos θ --> a/b < 1

What are the number of petals for roses when n is even in r = a cos nθ or r = a sin(nθ)?

2n petals

What are the number of petals for roses when n is odd in r = a cos nθ or r = a sin(nθ)?

n petals