Lesson 11Interpreting Graphs of Proportional Relationships

Let’s read stories from the graphs of proportional relationships.

Learning Targets:

  • I can draw the graph of a proportional relationship given a single point on the graph (other than the origin).
  • I can find the constant of proportionality from a graph.
  • I understand the information given by graphs of proportional relationships that are made of up of points or a line.

11.1 What Could the Graph Represent?

Here is a graph that represents a proportional relationship.

A line is graphed in the coordinate plane with the origin labeled “O”. The x axis has the numbers 0 through 20, in increments of 2, indicated. The y axis has the numbers 0 through 20, in increments of 2, indicated. The line begins at the origin. It moves gradually upward and to the right passing through the point with coordinates 8 comma 6. There is also a point indicated on the line and the coordinates of that point have integer values.
  1. Invent a situation that could be represented by this graph.
  2. Label the axes with the quantities in your situation.
  3. Give the graph a title.
  4. There is a point on the graph. What are its coordinates? What does it represent in your situation?

11.2 Tyler's Walk

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

  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?
  2. 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.
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
  1. What does the point (0, 0) mean in this situation?
  2. 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.
  1. 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?

11.3 Seagulls Eat What?

4 seagulls ate 10 pounds of garbage. Assume this information describes a proportional relationship.

  1. Plot a point that shows the number of seagulls and the amount of garbage they ate.
  2. Use a straight edge to draw a line through this point and (0,0) .
  3. Plot the point (1, k) on the line. What is the value of k ? What does the value of k tell you about this context?

Lesson 11 Summary

For the relationship represented in this table, y is proportional to x . We can see in the table that \frac54 is the constant of proportionality because it’s the y value when x is 1.

The equation y = \frac54 x 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 x 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 x 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 \frac54 is the y -coordinate of the point on the graph where the x -coordinate is 1. This could mean the snail is traveling \frac54 feet per minute or that the recipe calls for 1\frac14 cups of yogurt for every teaspoon of cinnamon.

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

Lesson 11 Practice Problems

  1. There is a proportional relationship between the number of months a person has had a streaming movie subscription and the total amount of money they have paid for the subscription. The cost for 6 months is $47.94. The point (6, 47.94) is shown on the graph below.

    A coordinate plane with the origin labeled “O”. The horizontal axis is labeled “time in months” and the numbers 0 through 22 are indicated. The vertical axis is labeled “cost in dollars” and the numbers 0 through 180, in increments of 10, are indicated. The point with coordinates 6 comma 47.94 is indicated.
    1. What is the constant of proportionality in this relationship?
    2. What does the constant of proportionality tell us about the situation?
    3. Add at least three more points to the graph and label them with their coordinates.
    4. Write an equation that represents the relationship between C , the total cost of the subscription, and m , the number of months.
  2. The graph shows the amounts of almonds, in grams, for different amounts of oats, in cups, in a granola mix. Label the point (1, k) on the graph, find the value of k , and explain its meaning.

  3. To make a friendship bracelet, some long strings are lined up; then, one string is tied in a knot with each of the other strings to create a row of knots. A new string is chosen and knotted with the all the other strings to create a second row. This process is repeated until there are enough rows to make a bracelet to fit around your friend's wrist.

    Are the number of knots proportional to the number of rows? Explain your reasoning.

  4. What information do you need to know to write an equation relating two quantities that have a proportional relationship?