8.10: Drive Away
Look at the fact and graph below. Can you figure out what Bush's speed was in miles per hour? Can you complete a table to show the total number of miles traveled by Bush at that speed? Can you write a rule to show how the number of miles is related to the number of hours? In this section, we will learn how to interpret graphs and write rules to match what we see on graphs.
Fact: The graph shows the speeds of drivers, Axel and Bush, on city roads. Bush is traveling faster than Axel.
Drive Away
In order to make a table and write a rule for situations like the one above, we can use the problem solving steps to help.
 First, describe what you know. What information do we see in the graph?
 Second, identify what your job is. In these problems, your job will be to make a table and write a rule.
 Third, make a plan. In these problems, your plan should be to use the graph to fill in the table. Then, look for a pattern to help you write the rule.
 Fourth, solve. Implement your plan.
 Fifth, check. Make sure your rule works with the graph.
1. Fact: The graph shows the speeds of drivers, Harold and French. Harold didn’t travel as fast as French.
 What was Harold’s speed in miles per hour?
 Complete a table to show total number of miles traveled by Harold at that speed.
 Let \begin{align*}t\end{align*}
t represent number of hours and \begin{align*}D\end{align*}D represent distance in number of miles. Write a rule to show how the number of miles \begin{align*}(D)\end{align*}(D) is related to number of hours \begin{align*}(t)\end{align*}(t) .
We can use problem solving steps to help us to interpret the graph and answer the questions.
\begin{align*}& \mathbf{Describe:} && \text{The graph shows time in number of hours along the horizontal axis.}\\ &&& \text{Distance in number of miles is shown along the vertical axis. The two}\\ &&& \text{lines on the graph represent the speeds of Harold and French. The Fact}\\ &&& \text{indicates that Harold didn't travel as fast as French.}\\ \\ & \mathbf{My \ Job:} && \text{Use the Fact to figure out which line represents Harold. Complete the table}\\ &&& \text{for Harold. Write a rule to describe how the number of miles traveled is}\\ &&& \text{related to the number of hours of driving time.}\\ \\ & \mathbf{Plan:} && \text{Compare the lines with the Fact and decide which line represents Harold. }\\ &&& \text{Determine his average speed. Use that data to complete the table.}\\ &&& \text{Generalize from the data in the table. Construct the rule.}\\ \\ & \mathbf{Solve:} && \text{Harold is line Q.}\\ &&& 1. \ \text{He was driving at a speed of} \ 15 \ \text{mph.}\\ &&& 2. \ \text{Complete the table using that speed to figure out the distances.}\end{align*}
\begin{align*}& 3. \ D=15 \ t\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\end{align*}
\begin{align*}& \mathbf{Check:} && \text{Use the rule to verify the data in the table.} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \\ &&& D=15 \times 1=15\\ &&& D=15 \times 2=30\\ &&& D=15 \times 3=45\\ &&& D=15 \times 4=60\\ &&& D=15 \times 5=75\\ &&& D=15 \times 6=90\end{align*}
2. Fact: The graph shows the speeds of drivers, Roberts and Clark. Roberts drove 10 mph slower than Clark.
 What was Clark’s speed in miles per hour?
 Complete a table to show total number of miles traveled by Clark at that speed.
 Let \begin{align*}t\end{align*} represent number of hours and \begin{align*}D\end{align*} represent distance in number of miles. Write a rule to show how the number of miles \begin{align*}(D)\end{align*} is related to number of hours \begin{align*}(t)\end{align*}.
We can use problem solving steps to help us to interpret the graph and answer the questions.
\begin{align*}& \mathbf{Describe:} && \text{The graph shows time in number of hours along the horizontal axis.}\\ &&& \text{Distance in number of miles is shown along the vertical axis. The two}\\ &&& \text{lines on the graph represent the speeds of Roberts and Clark. The Fact}\\ &&& \text{indicates that Roberts drove 10 mph slower than Clark.}\\ \\ & \mathbf{My \ Job:} && \text{Use the Fact to figure out which line represents Clark. Complete the table}\\ &&& \text{for Clark. Write a rule to describe how the number of miles traveled is}\\ &&& \text{related to the number of hours of driving time.}\\ \\ & \mathbf{Plan:} && \text{Compare the lines with the Fact and decide which line represents Clark. }\\ &&& \text{Determine his average speed. Use that data to complete the table.}\\ &&& \text{Generalize from the data in the table. Construct the rule.}\\ \\ & \mathbf{Solve:} && \text{Clark is line S.}\\ &&& 1. \ \text{He was driving at a speed of} \ 20 \ \text{mph.}\\ &&& 2. \ \text{Complete the table using that speed to figure out the distances.}\end{align*}
\begin{align*}& 3. \ D=20 \ t\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\end{align*}
\begin{align*}& \mathbf{Check:} && \text{Use the rule to verify the data in the table.} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \\ &&& D=20 \times 1=20\\ &&& D=20 \times 2=40\\ &&& D=20 \times 3=60\\ &&& D=20 \times 4=80\\ &&& D=20 \times 5=100\\ &&& D=20 \times 6=120\end{align*}
Earlier Problem Revisited
Fact: The graph shows the speeds of drivers, Axel and Bush, on city roads. Bush is traveling faster than Axel.
 What was Bush’s speed in miles per hour?
 Complete a table to show total number of miles traveled by Bush at that speed.
 Let \begin{align*}t\end{align*} represent number of hours and \begin{align*}D\end{align*} represent distance in number of miles. Write a rule to show how the number of miles \begin{align*}(D)\end{align*} is related to number of hours \begin{align*}(t)\end{align*}.
We can use problem solving steps to help us to interpret the graph and answer the questions.
\begin{align*}& \mathbf{Describe:} && \text{The graph shows time in number of hours along the horizontal axis.}\\ &&& \text{Distance in number of miles is shown along the vertical axis. The two}\\ &&& \text{lines on the graph represent the speeds of Axel and Bush. The Fact}\\ &&& \text{indicates that Bush is traveling faster than Axel.}\\ \\ & \mathbf{My \ Job:} && \text{Use the Fact to figure out which line represents Bush. Complete the table}\\ &&& \text{for Bush. Write a rule to describe how the number of miles traveled is}\\ &&& \text{related to the number of hours of driving time.}\\ \\ & \mathbf{Plan:} && \text{Compare the lines with the Fact and decide which line represents Bush. }\\ &&& \text{Determine his average speed. Use that data to complete the table.}\\ &&& \text{Generalize from the data in the table. Construct the rule.}\\ \\ & \mathbf{Solve:} && \text{Bush is line S.}\\ &&& 1. \ \text{He was driving at a speed of} \ 30 \ \text{mph.}\\ &&& 2. \ \text{Complete the table using that speed to figure out the distances.}\end{align*}
\begin{align*}& 3. \ D=30 \ t\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\end{align*}
\begin{align*}& \mathbf{Check:} && \text{Use the rule to verify the data in the table.} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \\ &&& D=30 \times 1=30\\ &&& D=30 \times 2=60\\ &&& D=30 \times 3=90\\ &&& D=30 \times 4=120\\ &&& D=30 \times 5=150\\ &&& D=30 \times 6=180\end{align*}
Vocabulary
A graph is one way to show the relationship between two variables. In this concept, we looked at graphs that showed the relationship between distance and time. A table is another way to show a relationship between two variables (often thought of as the input and the output). In this concept, the inputs of our tables were number of hours and the outputs of our tables were number of miles. A rule is an equation that can describe the relationship between the variables in a graph or a table. In this concept, we wrote rules that showed the relationship between the number of hours and the number of miles.
Examples
Example 1
Fact: The graph shows the speeds of drivers, Marx and Stevens. Marx drove 5 mph faster than Stevens.
 a. What was Marx’s speed in miles per hour?
 25 mph
 b. Complete a table to show total number of miles traveled by Marx at that speed.
 c. Let \begin{align*}t\end{align*} represent number of hours and \begin{align*}D\end{align*} represent distance in number of miles. Write a rule to show how the number of miles \begin{align*}(D)\end{align*} is related to number of hours \begin{align*}(t)\end{align*}.
 \begin{align*}D = 25 \ t\end{align*}
Example 2
Fact: The graph shows the speeds of drivers, Ellsworth and Stewart. The speed limit on Highway 999 is 55 mph. Ellsworth obeyed the speed limit. Stuart did not.
 a. What was Ellsworth’s speed in miles per hour?
 50 mph
 b. Complete a table to show total number of miles traveled by Ellsworth at that speed.

 \begin{align*}t\end{align*}\begin{align*}D\end{align*}\begin{align*}(D)\end{align*}\begin{align*}(t)\end{align*}. is related to number of hours represent distance in number of miles. Write a rule to show how the number of miles represent number of hours and c. Let
 \begin{align*}D = 50 \ t\end{align*}
Review
Fact: The graph below shows the speeds of drivers, Stacey and Kimmy. Kimmy drove 5 mph faster than Stacey.
 What was Kimmy’s speed in miles per hour?
 Complete a table to show total number of miles traveled by Kimmy at that speed.
 Let \begin{align*}t\end{align*} represent number of hours and \begin{align*}D\end{align*} represent distance in number of miles. Write a rule to show how the number of miles \begin{align*}(D)\end{align*} is related to number of hours \begin{align*}(t)\end{align*}.
 What was Stacey’s speed in miles per hour?
 Complete a table to show total number of miles traveled by Stacey at that speed.
 Let \begin{align*}t\end{align*} represent number of hours and \begin{align*}D\end{align*} represent distance in number of miles. Write a rule to show how the number of miles \begin{align*}(D)\end{align*} is related to number of hours \begin{align*}(t)\end{align*}.
Fact: The graph below shows the speeds of drivers, Wayne and Mike. Wayne drove 25 mph faster than Mike.
 What was Wayne’s speed in miles per hour?
 Complete a table to show total number of miles traveled by Wayne at that speed.
 Let \begin{align*}t\end{align*} represent number of hours and \begin{align*}D\end{align*} represent distance in number of miles. Write a rule to show how the number of miles \begin{align*}(D)\end{align*} is related to number of hours \begin{align*}(t)\end{align*}.
 What was Mike’s speed in miles per hour?
 Complete a table to show total number of miles traveled by Mike at that speed.
 Let \begin{align*}t\end{align*} represent number of hours and \begin{align*}D\end{align*} represent distance in number of miles. Write a rule to show how the number of miles \begin{align*}(D)\end{align*} is related to number of hours \begin{align*}(t)\end{align*}.
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Image Attributions
Students analyze graphs that show the relationship between time and distance. They make tables and write rules for the relationships shown in the graphs. Students use problem solving steps to help.
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