FoundationsConcept reference

Affine functions

A scalar affine function multiplies its input by a fixed slope and adds a fixed intercept.

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01A service uses 5 credits to start, plus 2 credits per job#

A service uses 5 credits to start, plus 2 credits per job. At 3 jobs, the job cost is $2\times3=6$ credits; adding the starting cost gives 11.

The affine rule records both parts.

f(x)=mx+b

$m$ is the slope, the output change per input unit. $b$ is the intercept, the output at $x=0$ . Here, $m=2$ and $b=5$ .

Check your reasoning

For $f(x)=2x+5$ , what is $f(4)$ ?

Show answer and explanation

Multiply 4 by 2, then add 5: $8+5=13$ .

At $x=0$ , the scaled part is zero and $f(0)=b$ . An intercept can be negative: with $m=3$ and $b=-2$ , $f(2)=3(2)-2=4$ .

Check your reasoning

Slope is 3; intercept is −2. Which expression gives $f(4)$ ?

Show answer and explanation

$3(4)-2$

Multiply by 3, then add the intercept −2.

Check your reasoning

A calibration rule has slope −2 and intercept 7. What output does it assign to input 3?

Show answer and explanation

Use $(-2)(3)+7=-6+7=1$ .

Key takeaway