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Precalculus Unit 1: Functions review & rates of change

Precalculus studies functions by how they change. Average rate of change, increasing and decreasing behavior, and concavity are the main tools.

Covers: Function notation, average rate of change, concavity in tables/graphs

Study guide

Precalculus begins by treating functions as processes that turn inputs into outputs, described with tables, graphs, formulas and words. Function notation f(x) names the output for input x, and you should be comfortable reading values, domains and ranges from any representation.

The average rate of change of f on [a, b] is [f(b) − f(a)]/(b − a): the slope of the secant line through the two points. It measures how fast the output changes per unit of input over that interval. A function is increasing where outputs rise as inputs rise and decreasing where they fall; relative maxima occur where it switches from increasing to decreasing, and relative minima the reverse.

Concavity describes how the rate of change itself is changing. If the rate of change is increasing, the graph is concave up (bending upward); if it's decreasing, the graph is concave down. In a table with equally spaced inputs, look at the differences between consecutive outputs: constant first differences mean the function is linear; constant second differences mean it's quadratic; first differences that grow mean concave up and that shrink mean concave down. A point where concavity changes is an inflection point.

On an interval where a function is concave up, its graph lies below the secant line, so an estimate made with the average rate of change overestimates the true value. Where it is concave down, the graph lies above the secant line and the estimate is too low. To find a domain from a formula, exclude inputs that make a denominator zero, that make the expression under an even root negative, or that make a logarithm's input zero or negative. For √(p(x)/q(x)), solve p(x)/q(x) ≥ 0 with a sign chart and leave out the zeros of q.

A piecewise function uses different formulas on different intervals of x. To evaluate it, find the interval that contains the input (watch whether each endpoint is included) and use only that piece's formula.

Key ideas

• Average rate of change on [a, b] is the slope of the secant line.
• A function is increasing where outputs rise as inputs rise.
• Concave up: rate of change is increasing. Concave down: rate of change is decreasing.
• In a table with equal input steps, constant first differences mean linear; constant second differences mean quadratic.
• Relative extrema happen where a function switches between increasing and decreasing.

Formulas and rules

Common mistakes

• Confusing increasing with concave up.
• Using second differences when the input steps aren't equal.

Words to know

Average rate of change on [a, b]

[f(b) − f(a)] / (b − a)

Constant first differences

The data are linear.

Constant second differences

The data are quadratic.

Concave down

The rate of change is decreasing.

Relative maximum

Where a function switches from increasing to decreasing.

Average rate of change

[f(b) − f(a)]/(b − a): the slope of the secant line.

Secant line

A line through two points on a graph.

Increasing function

Outputs rise as inputs rise.

Concave up

The rate of change is increasing; the graph bends upward.

Inflection point

Where concavity changes.

Relative minimum

Where a function switches from decreasing to increasing.

Sample practice questions

19 more questions like these, from level 1 to 5, are in the practice set.