Functions and graphs

A function is a rule that gives exactly one output for each input. Functions are the central idea of college algebra and of calculus. A graph lets us see a function: where it rises, where it falls, and where it crosses the axes.

In this chapter, you will learn to graph equations, work with lines and circles, and describe functions with formulas, tables, and graphs.

This chapter covers the following topics:

Coordinates, graphs, and circles

The coordinate plane, the distance and midpoint formulas, graphs of equations and their intercepts, symmetry, and the equation of a circle.

Lines

Slope, the slope-intercept and point-slope forms, horizontal and vertical lines, and parallel and perpendicular lines.

Functions

What a function is, function notation, domain and range, the vertical line test, the difference quotient, and piecewise functions.

Graphs of functions

Graphs of common functions, increasing and decreasing, even and odd functions, and shifting, stretching, and reflecting graphs.

Quadratic functions

Parabolas, the vertex form, the vertex formula, maximum and minimum values, and applications.

Combining functions

Sums, differences, products, and quotients of functions, composition of functions, and their domains.

Practice questions

Practice problems on every section of this chapter, with full step-by-step solutions.

Before you start

This chapter uses the skills from Chapter 2, especially solving quadratic equations, completing the square, and interval notation.