Math Core

In 7th grade you studied proportional relationships, like buying apples at a fixed price per pound. Their graphs are straight lines through the origin. But plenty of straight lines don't go through the origin. In this lesson you'll see how proportional relationships fit inside the bigger family of linear functions, and how to tell them apart.

Two lines with the same slope

Compare two ways to pay for bowling:

Plan A: $3 per game, no shoe rental. Cost: $y = 3x$ .
Plan B: $3 per game plus $2 for shoes. Cost: $y = 3x + 2$ .

Plan A (y = 3x) starts at the origin. Plan B (y = 3x + 2) starts 2 higher.Open in grapher →

Both lines have slope 3, since each extra game costs $3 either way. They are parallel. The difference is the starting point. Plan A starts at $0, so it goes through the origin. Plan B starts at $2.

Check the ratio $\dfrac{y}{x}$ for each plan:

games, $x$	Plan A, $y$	$\dfrac{y}{x}$	Plan B, $y$	$\dfrac{y}{x}$
1	3	3	5	5
2	6	3	8	4
4	12	3	14	3.5

Plan A has the same ratio every time, so it is proportional. Plan B's ratio keeps changing, so it is not proportional, even though it's still a straight line.

Proportional vs. linear

A linear function has the form $y = mx + b$ . Its graph is a straight line.
A proportional relationship is the special linear function with $b = 0$ : $y = mx$ . Its graph goes through the origin $(0, 0)$ .

Every proportional relationship is linear. A linear function is proportional only when $b = 0$ .

In a proportional relationship $y = kx$ , the slope $k$ is the unit rate (or constant of proportionality). For Plan A, the unit rate is $3 per game.

How to tell them apart

representation	proportional	linear, not proportional
equation	$y = mx$	$y = mx + b$ with $b \ne 0$
graph	straight line through $(0, 0)$	straight line that misses $(0, 0)$
table	$\dfrac{y}{x}$ always the same	$y$ changes steadily, but $\dfrac{y}{x}$ varies

Worked example: Which tables are proportional?

$x$	2	4	6
Table 1: $y$	5	10	15
Table 2: $y$	7	11	15

Table 1: $\dfrac{5}{2} = \dfrac{10}{4} = \dfrac{15}{6} = 2.5$ . Proportional: $y = 2.5x$ .

Table 2: $y$ goes up by 4 each time $x$ goes up by 2, so it's linear with slope 2. But $\dfrac{7}{2} = 3.5$ and $\dfrac{11}{4} = 2.75$ are different, so it's not proportional. Work backward to $x = 0$ : subtract 4 from 7 to get $y = 3$ . So $y = 2x + 3$ , and $b = 3$ .

Worked example: Reading a graph

A line passes through $(0, 0)$ and $(4, 6)$ . Is it proportional? What is its equation?

y = 3x/2(0, 0)(4, 6)Open in grapher →

It goes through the origin, so yes. The slope is $\dfrac{6}{4} = 1.5$ , so the equation is $y = 1.5x$ . The unit rate is 1.5.

Common mistake

A straight line is not automatically proportional. Always check whether it passes through $(0, 0)$ , or whether $b = 0$ in the equation.

Worked example: Comparing two plans

Phone plan X charges $0.25 per minute. Plan Y charges $15 per month plus $0.10 per minute. Write each cost $y$ for $x$ minutes and say which is proportional.

Plan X: $y = 0.25x$ . Proportional (0 minutes costs $0).
Plan Y: $y = 0.10x + 15$ . Linear but not proportional (0 minutes still costs $15).

For 60 minutes, X costs $0.25(60) = 15$ dollars and Y costs $0.10(60) + 15 = 21$ dollars.

Practice

Practice 1

Which equation shows a proportional relationship?

Practice 2

This table shows a linear function. What is its $y$ -intercept?

$x$	1	2	3
$y$	5	8	11

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

Practice 3

This line shows a proportional relationship. Write its equation.

y = 5x/2(0, 0)(2, 5)Open in grapher →

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 4

A pool already holds 200 gallons of water. A hose adds 50 gallons per minute. Write an equation for the gallons $y$ in the pool after $x$ minutes.

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 5

Which table is linear but not proportional?

Practice 6

A line has slope 4 and passes through $(1, 4)$ . Is it a proportional relationship?

Practice 7

At Store A, the cost of grapes is proportional to the weight: 3 pounds cost $7.50. Store B charges $2 per pound plus a $1.50 bag fee. How many dollars less does Store B charge for 5 pounds?

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