Lesson 7Graphs of Proportional Relationships

Learning Goal

Let’s see how graphs of proportional relationships differ from graphs of other relationships.

Learning Targets

I can find the constant of proportionality from a graph.
I know that the graph of a proportional relationship lies on a line through $(0, 0)$ .

Lesson Terms

coordinate plane
origin

Warm Up: Notice These Points

Plot the points.
$(0, 10), (1, 8), (2, 6), (3, 4), (4, 2)$
open applet in presentation mode
What do you notice about the graph?

Print Version

Plot the points $(0, 10), (1, 8), (2, 6), (3, 4), (4, 2)$ .
What do you notice about the graph?

Activity 1: T-shirts for Sale

Some T-shirts cost $8 each.

$x$	$y$
$1$	$8$
$2$	$16$
$3$	$24$
$4$	$32$
$5$	$40$
$6$	$48$

Problem 1

Use the table to answer these questions.

What does $x$ represent?
What does $y$ represent?
Is there a proportional relationship between $x$ and $y$ ?

Problem 2

Plot the pairs from the previously given table on the coordinate plane.

open applet in presentation mode

Print Version

Plot the pairs from the previously given table on the coordinate plane.

A coordinate plane from 0 to 6 on the x axis and 0 to 50 on the y axis

Problem 3

What do you notice about the graph?

Activity 2: Tyler’s Walk

Tyler was at the amusement park. He walked at a steady pace from the ticket booth to the bumper cars.

A coordinate plane with the origin labeled “O”. The horizontal axis is labeled “elapsed time in seconds” and the numbers 0 through 60, in increments of 10, are indicated. There are vertical gridlines midway between. The vertical axis is labeled “distance from the ticket booth in meters” and the numbers 0 through 60, in increments of 10, are indicated. There are horizontal gridlines midway between. The point with coordinates 40 comma 50 is indicated.

time (seconds)	distance (meters)
$0$	$0$
$20$	$25$
$30$	$37.5$
$40$	$50$
$1$

The point on the graph shows his arrival at the bumper cars. What do the coordinates of the point tell us about the situation?
The table representing Tyler’s walk shows other values of time and distance. Complete the table. Next, plot the pairs of values on the grid.
What does the point $(0, 0)$ mean in this situation?
How far away from the ticket booth was Tyler after 1 second? Label the point on the graph that shows this information with its coordinates.
What is the constant of proportionality for the relationship between time and distance? What does it tell you about Tyler’s walk? Where do you see it in the graph?

Are you ready for more?

If Tyler wanted to get to the bumper cars in half the time, how would the graph representing his walk change? How would the table change? What about the constant of proportionality?

Lesson Summary

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.”

A graph of weight in pounds vs cost in dollars starting at (0,0) with points: (1,6), (2,12), (3,18), (4.5, 27).

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. We could buy, for example, 4.5 pounds of blueberries or 1.875 pounds of blueberries, so all the points in between the whole numbers make sense in the situation, so any point on the line is meaningful.

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:

There are points that satisfy the relationship lie on a straight line.
The line that they lie on passes through the origin, $(0, 0)$ .

Here are some graphs that do not represent proportional relationships:

Seven points plotted in the coordinate plane with the origin labeled “O”. The x axis has the numbers 0 through 7 indicated. The y axis has the numbers 0 through 6 indicated. The points with coordinates 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 are indicated.

These points do not lie on a line.

A graph with a line starting at (0, 2). The x axis is from 0 to 7 and the y axis is from 0 to 6

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

Here is a different example of a relationship represented by this table where $y$ is proportional to $x$ . We can see in the table that $\frac{5}{4}$ is the constant of proportionality because it’s the $y$ value when $x$ is 1.

The equation $y = \frac{5}{4}$ also represents this relationship.

$x$	$y$
$4$	$5$
$5$	$\frac{25}{4}$
$8$	$10$
$1$	$\frac{5}{4}$

Here is the graph of this relationship.

A line is graphed in the coordinate plane with the origin labeled “O”. The numbers 0 through 10 are indicated on the x axis. The numbers 0 through 10 are indicated on they axis. The line begins at the origin. It moves steadily upward and to the right passing through the points with coordinates 1 comma five-fourths, 4 comma 5, 5 comma 25 fourths, and 8 comma 10.

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

If $y$ represents the cups of yogurt and represents the teaspoons of cinnamon in a recipe for fruit dip, then the point $(4, 5)$ 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, because $\frac{5}{4}$ is the $y$ -coordinate of the point on the graph where the $x$ -coordinate is 1. This could mean the snail is traveling $\frac{5}{4}$ feet per minute or that the recipe calls for $1 \frac{1}{4}$ cups of yogurt for every teaspoon of cinnamon.

In general, when is proportional to $x$ , the corresponding constant of proportionality is the $y$ -value when $x = 1$ .