Linear Functions

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Idea

A rule $f(x)=mx+b$ has constant rate of change $m$ and vertical intercept $b$ . In strict linear-algebra terminology, a map with $b\ne0$ is affine; in elementary graphing these are commonly called linear functions.

Method

For distinct inputs,

m=\frac{f(x_2)-f(x_1)}{x_2-x_1}.

Positive $m$ means increasing, negative $m$ decreasing, and zero $m$ constant. A vertical line is not the graph of a function $y=f(x)$ .

Worked example

A line through $(1,3)$ and $(4,9)$ has slope $(9-3)/(4-1)=2$ . Substituting $(1,3)$ into $y=2x+b$ gives $b=1$ . Hence $f(x)=2x+1$ and increasing $x$ by $0.5$ increases the output by $1$ .

Common mistake

The intercept is the output at $x=0$ , not the input at which the graph crosses the horizontal axis.

Check your understanding

Find the line through $(0,4)$ and $(2,0)$ .

Show answer

$f(x)=-2x+4$ .

Explore

A line through two points

Try this. Move the two points and compare the equation. Equal x values produce a vertical line; equal points cannot define one.

Point A · x-2Point A · y3Point B · x3Point B · y1

y = -0.4x + 2.2 · Slope = -0.4

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

For f(x)=−2x+5, how much does the output change when x increases by 1?

−2

Hint 1 · Find a starting point

The coefficient of x is the rate of change.

Hint 2 · Take the next step

Compare f(x+1) and f(x).

Show the reasoning

Answer: −2

f(x+1)−f(x)=−2, independent of the initial input.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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