Day #7 - Proportional Relationships and Graphs
Essential question: How can you use graphs to represent and analyze proportional
relationships?
Example 1 – Graphing Proportional Relationships
Most showerheads that were manufactured before 1994 use 5 gallons of water per minute. Is
the relationship between the number of gallons of water and the number of minutes a
proportional relationship?
A) Complete the table.
Time (min)
1
Water Used (gal)
5
2
3
10
35
B) Based on the table, is this a proportional relationship? Explain your answer.
C) Plot the data from the table.
D) If you continued the table to include 23 minutes, would the point (23, 125) be on this
graph? Why or why not?
E) If a line was drawn through the plotted points, does it make sense that it would go
through the origin? Explain.
F) Another showerhead uses less water per minute. How would its graph compare to the
one you plotted?
In addition to using a table to determine if a relationship is proportional, you also can
use a graph.
A relationship is a __________________________ relationship if its graph is a
_______________________________ through the __________________.
Example 2 – Identifying Proportional Relationships
An Internet café charges a one-time $5 service fee and then $2 for every hour of use. Is
this relationship a proportional relationship?
A) Complete the table.
Time (h)
1
Total Cost ($)
7
2
5
8
17
B) Plot the data from the table and connect the points with a line.
C) The graph of the data is a ______________________________________________.
The line does/does not go through the origin.
So, the relationship is ___________________________________________________.
Now You Try It!
1) Plot the data from the table and connect the points with a line.
Canoe Rental (h)
2
5
8
10
Total Cost ($)
5 11 17 21
2) Is this a proportional relationship? Explain.
You have seen that the equation of a proportional relationship may be written as y = ax,
where a is a positive number. The constant of proportionality, a, tells you how steep the
graph of the relationship is. The greater the value of a, the steeper the line.
Example 3 – Analyzing Graphs
The graph shows the relationship between time in years and the number of centimeters a
fingernail grows.
A) What does the point (3, 9) represent?
B) What is the constant of proportionality?
C) Write an equation for the relationship.
D) What does the point (0, 0) on the graph represent?
E) What is the rate at which a fingernail grows? How does this relate to the constant of
proportionality?
Practice:
A) Complete each table. B) Tell whether the relationship is a proportional relationship.
Explain why or why not.
1. A student reads 65 pages per
hour.
Time
(h)
Pages
3
5
10
585
2. A babysitter makes $7.50 per
hour.
Time (h)
Earnings
2
5
22.50
60
5. A train travels at 72 miles per hour. Will the graph of the train's rate of speed show that
the relationship between the number of miles traveled and the number of hours is a
proportional relationship? Explain.
The graph shows the relationship between time and the distance run by two horses.
6. How long does it take each horse to run 1 mile?
7. What does the point (0, 0) represent?
8. Write an equation for the relationship between time and distance.
9. Draw a line on the graph representing a horse that is faster than each of these.