One way to represent a proportional relationship is with a graph. Here is a graph that represents different amounts that fit the situation, “Blueberries cost $6 per pound.” ​​​​​​

Different points on the graph tell us, for example, that 2 pounds of blueberries cost $12, and 4.5 pounds of blueberries cost $27.

Sometimes it makes sense to connect the points with a line, and sometimes it doesn’t. For example, we could buy 4.5 pounds of blueberries or 1.875 pounds, or any other part of a whole pound. So all the points between the whole numbers make sense in the situation, and any point on the line is meaningful.

Line graph. Weight in pounds. Cost in dollars. Horizontal axis, 0 to 5, by 1's. Vertical Axis, 0 to 40, by 10's. Line begins at origin, trends upward and right, passes through 1 comma 6, 2 comma 12, 3 comma 18, 4 point 5 comma 27.

If the graph represented the cost for different numbers of sandwiches (instead of pounds of blueberries), it might not make sense to connect the points with a line, because it is often not possible to buy 4.5 sandwiches or 1.875 sandwiches. Even if only points make sense in the situation, though, sometimes we connect them with a line anyway to make the relationship easier to see.

Graphs that represent proportional relationships all have a few things in common:

• Points that satisfy the relationship lie on a straight line.
• The line that they lie on passes through the origin, .

Here are some graphs that do not represent proportional relationships:

Graph of a non-proportional relationship, x y plane, origin O. Horizontal axis scale 0 to 7 by 1’s. Vertical axis scale 0 to 6 by 1’s. There are points at: (0 comma 0), (1 comma 1), (2 comma 3), (3 comma 4), (4 comma 4 point 5), (5 comma 5), (6 comma 5 point 1), and (7 comma 5 point 2).

These points do not lie on a line.

Line graph. Horizontal axis, 0 to 7, by 1's. Vertical Axis, 0 to 6, by 1's. Line begins on y axis at 0 comma 2, trends upward and right, passes through 2 comma 3, 4 comma 4, 6 comma 5.

This is a line, but it doesn’t go through the origin.

Here is a different example.

Here is the graph of this relationship.

Graph of a lin, x y plane, origin O. Horizontal and vertical axis scale 0 to 10 by 1’s. Line starts at (0 comma 0), rises to point (1 comma the fraction 5 over 4), rises to point (4 comma 5), rises to point (5 comma the fraction 25 over 4), then rises to point (8 comma 10) and keeps rising.

If represents the distance in feet that a snail crawls in minutes, then the point tells us that the snail can crawl 5 feet in 4 minutes.

If represents the cups of yogurt and represents the teaspoons of cinnamon in a recipe for fruit dip, then the point tells us that you can mix 4 teaspoons of cinnamon with 5 cups of yogurt to make this fruit dip.

We can find the constant of proportionality by looking at the graph:  is the -coordinate of the point on the graph where the -coordinate is 1. This could mean the snail is traveling feet per minute or that the recipe calls for cups of yogurt for every teaspoon of cinnamon.

In general, when is proportional to , the corresponding constant of proportionality is the -value when .