Chapter 04 of 10

Linear Functions

How can one constant rate be represented four different ways?

A linear function changes at a constant rate. You can recognize the same relationship in a situation, table, graph, or equation. Moving fluently among those representations is more important than memorizing any one form.

01

The big ideas

Understand these before you worry about speed.

Slope is rate of change

Slope compares vertical change to horizontal change. Positive slopes rise, negative slopes fall, zero slopes are horizontal, and vertical lines have undefined slope.

ExampleFrom (1, 2) to (5, 10), m=10−25−1=2\displaystyle m=\frac{10-2}{5-1}=2.

Choose a useful form

Slope-intercept form shows slope and y-intercept; point-slope form uses a known point; standard form often helps with intercepts.

Exampley = 2x − 3 has slope 2 and y-intercept −3.

Functions pair inputs with outputs

Each permitted input has exactly one output. Function notation f(x) names the output produced by input x.

ExampleIf f(x) = 3x + 1, then f(4) = 13.
02

Worked example

Follow the reason for each step—not just the symbols.

Problem

Write the equation of the line through (2, −1) and (6, 7).

  1. Find slope: m=7−(−1)6−2=84=2\displaystyle m=\frac{7-(-1)}{6-2}=\frac{8}{4}=2.
  2. Use point-slope form: y − (−1) = 2(x − 2).
  3. Simplify: y + 1 = 2x − 4, so y = 2x − 5.
Answery = 2x − 5
03

Try it yourself

Work on paper first. Open each answer only when you are ready to check.

1

Find the slope through (−1, 4) and (3, 12).

Check answerm = 2
2

Identify slope and y-intercept in y = −3x + 8.

Check answerSlope −3; y-intercept 8
3

Write a line with slope 12\displaystyle \frac{1}{2} through (4, 3).

Check answery=12x+1\displaystyle y=\frac{1}{2}x+1
4

Evaluate f(−2) for f(x) = x² + 3x.

Check answer−2
04

Practice and tools

Use the resource that matches what you need next.

Calculator

Slope Calculator

Find and interpret slope between two points.

Open resource Calculator

Point-Slope Calculator

Build an equation from two points step by step.

Open resource Worksheet

Graphing Lines

Graph lines in several forms on coordinate planes.

Open resource Worksheet

Writing Equations from Graphs

Move from visual information to equations.

Open resource Worksheet

Function Notation

Evaluate linear and nonlinear function rules.

Open resource