TransferCredit.org Guide to AP® Precalculus

This unit is 30–40% of the multiple-choice section. It builds the language of change, such as rates of change and concavity, then uses it on polynomial and rational functions, transformations and models. Every later unit reuses these ideas.

Key terms, with examples

function

A relation that maps each input value to exactly one output value. Two functions are equal only if they have the same domain and give the same outputs.

Example: Each second t after a ball is thrown gives one height h(t); two heights for one t would break the rule.

polynomial function

A function equivalent to p(x) = aₙxⁿ + … + a₂x² + a₁x + a₀, with n a positive integer, real coefficients and aₙ ≠ 0. A nonzero constant has degree zero.

Example: p(x) = 4x³ − x + 7 is a polynomial; g(x) = 2/x + 1 is not, since 2/x = 2x⁻¹.

rational function

A quotient of two polynomial functions, r(x) = p(x)/q(x). It measures the size of the numerator relative to the denominator at each input in its domain.

Example: r(x) = (x + 1)/(x² − 4) is rational; its domain leaves out x = 2 and x = −2.

parent function

The simplest function of a type, such as y = x² or y = √x. Transformations of it build models, and can change its domain and range.

Example: g(x) = √(x − 4) is the parent √x moved right 4, so its domain becomes x ≥ 4.

choosing a function model

Roughly constant rates of change suggest linear; roughly linear rates of change, or one symmetric maximum or minimum, suggest quadratic; several zeros or extrema suggest a higher-degree polynomial.

Example: Any five data points with different inputs can be fit exactly by a polynomial of degree 4 or less.

domain

The set of input values of a function. Unless told otherwise, it is every real x for which f(x) is a real number.

Example: f(x) = √(x − 3) has domain x ≥ 3, because square roots of negative numbers aren't real.

degree of a polynomial

The greatest exponent n with a nonzero coefficient. It gives the number of complex zeros and, with the sign of the leading coefficient, the end behavior.

Example: p(x) = 5 − 2x + 3x⁴ has degree 4, even though its terms are written in increasing order.

quotient of leading terms

For inputs of large magnitude, a rational function behaves like the numerator's leading term divided by the denominator's leading term.

Example: (6x³ + x)/(2x² − 5) behaves like 6x³/(2x²) = 3x, so it increases without bound as x → ∞.

additive transformation

Adding a constant to the output, f(x) + k, or to the input, f(x + h). It translates the graph vertically or horizontally without changing its shape.

Example: g(x) = f(x − 2) + 1 slides every point of the graph of f right 2 and up 1.

area and volume models

Contexts involving area, or two dimensions, are often modeled by quadratic functions. Contexts involving volume, or three dimensions, are often modeled by cubic functions.

Example: A square pen with side s meters has area s², a quadratic; a cube-shaped tank holds s³ cubic meters.

range

The set of output values a function actually produces over its domain.

Example: f(x) = x² + 1 has range y ≥ 1: no input gives an output below 1.

leading term and leading coefficient

In a polynomial of degree n, the leading term is aₙxⁿ and the leading coefficient is aₙ, in whatever order the terms are written.

Example: p(x) = 6x − x³ + 2: leading term −x³, leading coefficient −1.

What this area covers

How the exam asks it

Expect a function given by a formula, a short table, a described graph or a real context, then a question about where it increases, its concavity, its zeros, its end behavior, its asymptotes and holes, or a transformation of it. Many choices pair a claim with a reason, often in limit notation. Modeling questions ask you to pick a function type for data, use the units of a rate, or name an assumption or restriction of a model.

Common traps

Trap 1. Changes inside the parentheses act on x the opposite way: f(x + 3) moves the graph left 3, and f(2x) halves the x-coordinates rather than doubling them.

Trap 2. Factor and compare multiplicities before naming asymptotes. A factor that cancels completely leaves a hole, which is neither a vertical asymptote nor a zero.

Trap 3. Concavity follows the rate of change, not the outputs. A function can rise the whole time and still be concave down, rising more and more slowly.

Memory trick

Horizontal asymptote by degrees, "BOB-0, BOT-none, EATS-DC": Bigger On Bottom, y = 0; Bigger On Top, no horizontal asymptote; Exponents Are The Same, Divide the Coefficients.

Even multiplicity bounces off the x-axis; odd multiplicity goes through.