Math Core

A table shows a few pairs of values. A graph shows the whole relationship at once, and proportional relationships have a graph you can spot instantly. In this lesson you'll learn what that graph looks like and how to read the constant of proportionality right off it.

Plotting a proportional relationship

Tickets to a school play cost $3 each. Here is a table of the total cost:

tickets, $x$	cost in dollars, $y$
0	0
1	3
2	6
3	9
4	12

Plot each pair $(x, y)$ as a point.

Tickets (across) and total cost in dollars (up). The points lie on a line through the origin.Open in grapher →

The points line up perfectly, and the line passes through $(0, 0)$ , the origin. That's no accident: zero tickets cost zero dollars, and each extra ticket adds the same $3.

The graph of a proportional relationship

The graph of a proportional relationship is a straight line through the origin $(0, 0)$ .

If a graph is not a straight line, the relationship is not proportional.
If a straight line does not pass through $(0, 0)$ , the relationship is not proportional.

What doesn't count

Look at these two graphs. Neither one is proportional.

A is a straight line that crosses the y-axis at 2, not at 0. B passes through the origin but curves.Open in grapher →

Line A is straight, but it starts at $(0, 2)$ . Maybe it shows a cost with a $2 fee. When $x = 1$ , $\dfrac{y}{x} = 3$ , but when $x = 2$ , $\dfrac{y}{x} = 2$ . The ratio changes.
Graph B goes through the origin, but it bends. Its ratio $\dfrac{y}{x}$ keeps growing.

Common mistake

"Passes through the origin" and "is a straight line" are both required. Checking only one of them is the most common mistake. A curve through $(0, 0)$ is not proportional, and neither is a straight line that misses $(0, 0)$ .

Reading k from a graph

Every point $(x, y)$ on the line has the same ratio $\dfrac{y}{x}$ , which is the constant of proportionality $k$ . So you can pick any point on the line (other than the origin) and divide.

The point where $x = 1$ is especially useful. Its $y$ -value is the amount for 1 unit, which is the unit rate. So the line always passes through $(1, k)$ .

Worked example: Finding k from a point

The graph shows the gallons of water in a tank as it fills.

Minutes (across) and gallons of water (up).Open in grapher →

Find the constant of proportionality and explain what it means.

Use the point $(4, 10)$ : $k = \dfrac{10}{4} = 2.5$ . Check with $(2, 5)$ : $\dfrac{5}{2} = 2.5$ . The tank fills at 2.5 gallons per minute, so the line passes through $(1, 2.5)$ .

Worked example: What a point means

A graph of a proportional relationship shows hours worked $x$ and dollars earned $y$ . It passes through $(5, 60)$ . What do the points $(5, 60)$ , $(0, 0)$ and $(1, k)$ mean?

$(5, 60)$ : in 5 hours, you earn $60.
$(0, 0)$ : in 0 hours, you earn $0.
$k = \dfrac{60}{5} = 12$ , so the line passes through $(1, 12)$ : in 1 hour, you earn $12. That's the hourly pay rate.

Comparing two graphs

When two proportional graphs share the same axes, the steeper line has the greater constant of proportionality.

Worked example: Two hoses

Two hoses fill pools. The graph shows gallons (up) after some minutes (across).

Hose A passes through (2, 10). Hose B passes through (2, 6).Open in grapher →

Which hose is faster, and by how much?

Hose A: $k = \dfrac{10}{2} = 5$ gallons per minute.
Hose B: $k = \dfrac{6}{2} = 3$ gallons per minute.

Hose A's line is steeper, and it fills $5 - 3 = 2$ more gallons per minute.

Tip

To get $k$ quickly, look for a point on the line where both coordinates are easy to read, ideally where the line crosses grid corners exactly. Estimating between grid lines leads to small errors.

Practice

Practice 1

What is the constant of proportionality for this graph?

y = 4x(1, 4)(2, 8)(3, 12)Open in grapher →

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

Practice 2

What is the constant of proportionality for this graph?

y = 2x/3(3, 2)(6, 4)Open in grapher →

Type your answer like 2/5.

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

Practice 3

Which graph shows a proportional relationship?

y = x + 3y = 1.2xy = 2x - 3Open in grapher →

Practice 4

A graph of a proportional relationship shows minutes $x$ and gallons of gas pumped $y$ . The point $(8, 12)$ is on the graph. What does this point mean?

Practice 5

The graph shows a proportional relationship. What is $y$ when $x = 6$ ?

y = 3.5x(2, 7)Open in grapher →

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

Practice 6

Two cyclists ride at steady speeds. The graph shows distance in kilometers (up) and time in hours (across). How many more kilometers per hour does the faster cyclist ride?

y = 15xy = 12x(2, 30)(3, 36)Open in grapher →

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

Practice 7

The graph of a proportional relationship passes through $(5, 17.5)$ . It also passes through the point $(1, r)$ . What is $r$ ?

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

Practice 8

The graph of a proportional relationship passes through $(4, 6)$ . Which other point is on the graph?