AP Precalculus — Functions and Models flashcards

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Units 1–2 of AP Precalculus: rates of change, polynomial and rational functions, zeros, asymptotes and holes, sequences, exponential and logarithmic functions, inverses, solving equations, residuals, regression and semi-log plots.

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Learn deck · part of AP Precalculus · written against: College Board AP Precalculus Course and Exam Description (Units 1–3 (exam-assessed units)). Updated 2026-10-10.

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What does it mean for a function to be increasing on an interval, stated in terms of inputs and outputs?

For any two inputs $a < b$ in the interval, $f(a) < f(b)$ ⁠: as input values increase, output values increase.
Polynomial and Rational Functions › Rates of Change
How do you compute and interpret the average rate of change of $f$ from $x = a$ to $x = b$ ⁠?

$\frac{f(b) - f(a)}{b - a}$ ⁠, the slope of the secant line through the two points, measured in output units per input unit.
Polynomial and Rational Functions › Rates of Change
How can you tell from average rates of change that a graph is concave down on an interval?

Over consecutive equal-length input intervals, the average rates of change are decreasing. Increasing average rates of change mean concave up.
Polynomial and Rational Functions › Rates of Change
How do linear and quadratic functions differ in their rates of change over equal-length input intervals?

A linear function has a constant rate of change; for a quadratic, the average rates of change themselves change by a constant amount, so the second differences in a table are constant.
Polynomial and Rational Functions › Rates of Change
In precalculus terms, where does a function have a point of inflection?

Where the rate of change switches from increasing to decreasing, or the reverse, which is where the graph changes concavity.
Polynomial and Rational Functions › Rates of Change
How do arithmetic and geometric sequences differ, and what are their general terms?

Arithmetic: constant difference $d$ ⁠, $a_n = a_0 + dn$ ⁠. Geometric: constant ratio $r$ ⁠, $g_n = g_0 r^n$ ⁠.
Exponential and Logarithmic Functions › Sequences
An arithmetic sequence has $a_3 = 11$ and $a_7 = 27$ ⁠. What is $a_n$ ⁠?

Four steps add 16, so $d = 4$ and $a_0 = 11 - 12 = -1$ ⁠: $a_n = -1 + 4n$ ⁠.
Exponential and Logarithmic Functions › Sequences
Why can an exponential function be viewed as a geometric sequence with a continuous domain?

In $f(x) = ab^x$ ⁠, each increase of 1 in $x$ multiplies the output by $b$ ⁠, just as each term of a geometric sequence is the previous term times $r$ ⁠.
Exponential and Logarithmic Functions › Sequences
What are the domain, range and horizontal asymptote of $f(x) = ab^x$ with $a > 0$ ⁠, $b > 0$ ⁠, $b \neq 1$ ⁠?

Domain all real numbers, range $y > 0$ ⁠, asymptote $y = 0$ ⁠. It is increasing when $b > 1$ and decreasing when $0 < b < 1$ ⁠.
Exponential and Logarithmic Functions › Exponential Functions
Rewrite $f(x) = 5 \cdot 2^{x+3}$ without a horizontal shift.

$2^{x+3} = 2^3 \cdot 2^x$ ⁠, so $f(x) = 40 \cdot 2^x$ ⁠: for exponentials a horizontal translation is the same as a vertical dilation.
Exponential and Logarithmic Functions › Exponential Functions

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Topic outline follows the College Board AP Precalculus Course and Exam Description (effective Fall 2026), Units 1–2 (the AP name is used nominatively; no course or exam text reproduced).
OpenStax Precalculus 2e and OpenStax Calculus Volume 1, consulted as references; no text copied.

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