Math Core

Functions show up in many disguises: an equation, a table, a graph, or a sentence. To compare two of them, like two savings plans or two phone plans, you need to pull the same facts out of each one, no matter how it's written.

Two key numbers

For a linear function $y = mx + b$ , two numbers tell the story:

The rate of change $m$ (the slope): how much $y$ changes when $x$ goes up by 1.
The initial value $b$ (the $y$ -intercept): the value of $y$ when $x = 0$ .

Finding rate of change and initial value

representation	rate of change	initial value
equation $y = mx + b$	$m$	$b$
table	$\dfrac{\text{change in } y}{\text{change in } x}$	the $y$ -value when $x = 0$
graph	rise over run	where the line crosses the $y$ -axis
words	"per", "each", "every"	"starts at", "fee", "already has"

Worked example: An equation vs. a table

Function A is $y = 4x + 1$ . Function B is given by this table.

$x$	0	2	4
$y$	3	11	19

Compare their rates of change and initial values.

Function A: rate of change 4, initial value 1.
Function B: $y$ rises 8 when $x$ rises 2, so the rate is $\dfrac{8}{2} = 4$ . At $x = 0$ , $y = 3$ , so the initial value is 3.

The rates are equal, so the lines are parallel. Function B always stays 2 units above Function A.

Worked example: Words vs. a graph

Jon has $40 saved and adds $15 each week. Lia's savings are shown on the graph. Who saves faster, and who started with more?

Lia's savings in dollars after x weeks.Open in grapher →

Jon: $y = 15x + 40$ . Rate $15 per week, start $40.
Lia: rate $\dfrac{90 - 10}{4 - 0} = \dfrac{80}{4} = 20$ dollars per week, start $10.

Lia saves faster ($20 vs. $15 a week). Jon started with more ($40 vs. $10).

Tip

When comparing "which grows faster," compare rates of change. When comparing "who started ahead," compare initial values. These can have different winners, as with Jon and Lia.

Linear or nonlinear?

A linear function changes by the same amount every time $x$ goes up by the same step. Not every function does.

Look at $y = x^2$ , the area of a square with side $x$ :

$x$	0	1	2	3	4
$y$	0	1	4	9	16

The changes in $y$ are $1, 3, 5, 7$ . They aren't constant, so the function is nonlinear. Its graph curves.

y = x² curves upward. The line y = 2x + 1 is straight.Open in grapher →

Spotting a linear function

Equation: it can be written as $y = mx + b$ . No $x^2$ , no $x$ in a denominator, no $x$ as an exponent.
Table: equal steps in $x$ give equal changes in $y$ .
Graph: a straight line.

Worked example: Classify each function

$y = 3x - 5$
$y = x^2 + 1$
$y = \dfrac{6}{x}$
$y = \dfrac{x}{6}$

Solutions.

Linear: slope 3, intercept $-5$ .
Nonlinear: it has $x^2$ .
Nonlinear: $x$ is in the denominator. Try $x = 1, 2, 3$ : $y = 6, 3, 2$ . The changes are $-3$ and $-1$ .
Linear: $\dfrac{x}{6} = \dfrac{1}{6}x$ , a line with slope $\dfrac{1}{6}$ .

Common mistake

Don't confuse $y = \dfrac{x}{6}$ with $y = \dfrac{6}{x}$ . Dividing $x$ by a number is linear. Dividing a number by $x$ is not.

Practice

Practice 1

Function A is $y = 3x + 7$ . Function B is given by the table. Which has the greater rate of change?

$x$	1	2	3
$y$	5	10	15

Practice 2

Function A is shown on the graph. Function B is $y = 2x + 5$ . How much greater is Function B's initial value than Function A's?

y = 2x - 2(0, -2)(3, 4)Open in grapher →

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

Which function is nonlinear?

Practice 4

Is the function in this table linear?

$x$	0	1	2	3
$y$	1	2	4	8

Practice 5

Printer A prints 90 pages in 3 minutes at a steady rate. Printer B's pages are shown on the graph. How many more pages per minute does Printer A print?

Printer B: pages printed after x minutes.Open in grapher →

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

Function A is given by the table. Function B has the same rate of change as Function A, but its initial value is 8. When $x = 10$ , how much greater is Function B than Function A?

$x$	2	4	6
$y$	11	17	23

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

Which situation is described by a nonlinear function?