9.8: Pedal Power
Read the fact and analyze the graph below. Can you complete a table for each biker showing the distance they traveled? Can you write a function to show the relationship between the number of miles traveled \begin{align*}(D)\end{align*} and number of hours \begin{align*}(t)\end{align*} traveled for each biker? In this concept, we will practice writing functions from graphs.
Fact: Judson left on the bike hike one hour after Connor and biked at a faster speed than Connor.
Guidance
In order to make a table and write a function 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 function.
- 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 function.
- Fourth, solve. Implement your plan.
- Fifth, check. Make sure your function works with the graph.
Example A
Fact: Spiro left 1 hour after Hendricks and biked at a slower speed.
Use the Fact and the graph. Complete the table for each biker showing distance traveled. Write a function to show the relationship between number of miles traveled \begin{align*}(D)\end{align*} and number of hours \begin{align*}(t)\end{align*} traveled for each biker.
Solution:
We can use problem solving steps to help us to analyze the graph, fill in the table, and write a function.
\begin{align*}& \mathbf{Describe:} && \text{The graph is a Distance-Time graph and shows two lines. One line is} \\ & && \text{steeper than the other. One line starts where}\ t = 0.\ \text{The other line starts} \\ & && \text{where}\ t = 1.\ \text{The Fact identifies two bikers, Spiro and Hendricks}. \\ & && \text{Spiro left 1 hour after Hendricks and biked at a slower speed.} \\ \\ & \mathbf{My \ Job:} && \text{Use the graph and the Fact to figure out which line represents each biker}. \\ & && \text{Use the graph data to complete the table for each biker. Write the function} \\ & && \text{that shows the relationship between Distance and Time for each biker}. \\ \\ & \mathbf{Plan:} && \text{Use the Fact and graph to identify the line that represents Spiro} \\ & && \text{Complete the table for Spiro using points on the line. Write the} \\ & && \text{function rule. Do the same for Hendricks}. \\ \\ & \mathbf{Solve:} && \text{Since Spiro leaves later than Hendricks, the green line represents Spiro. Spiro's} \\ & && \text{number of hours is one less than the number of hours Hendricks bikes}. \\ & && \text{Hendricks' line is steeper. This means that Hendricks is biking faster than Spiro}.\end{align*}
Time (Number of hours) | Spiro Distance (Number of miles) | Hendricks Distance (Number of miles) |
---|---|---|
0 | 0 | 0 |
1 | 0 | 6 |
2 | 3 | 12 |
3 | 6 | 18 |
4 | 9 | 24 |
5 | 12 | 30 |
\begin{align*}& && \text{Spiro}:\ D = 3(t - 1) \\ & && \text{Hendricks}:\ D = 6t\end{align*}
\begin{align*}& \mathbf{Check:} && \text{Replace variables in the functions with their values. Check results}\\ & &&\text{with data in the table}.\end{align*}
Spiro \begin{align*}D = 3(t - 1)\end{align*} | Hendricks \begin{align*}D = 6t\end{align*} | |
---|---|---|
For \begin{align*}t = 1\end{align*} | \begin{align*}0=3(1-1)\end{align*} | \begin{align*}6 = 6 \times 1\end{align*} |
For \begin{align*}t = 2\end{align*} | \begin{align*}3 = 3(2 - 1)\end{align*} | \begin{align*}12 = 6 \times 2\end{align*} |
For \begin{align*}t = 3\end{align*} | \begin{align*}6 = 3(3 - 1)\end{align*} | \begin{align*}18 = 6 \times 3\end{align*} |
For \begin{align*}t = 4\end{align*} | \begin{align*}9 = 3(4 - 1)\end{align*} | \begin{align*}24 = 6 \times 4\end{align*} |
For \begin{align*}t = 5\end{align*} | \begin{align*}12 = 3(5 - 1)\end{align*} | \begin{align*}30 = 6 \times 5\end{align*} |
Example B
Fact: Kelly left 2 hours after Finley and biked at a faster speed.
Use the Fact and the graph. Complete the table for each biker showing distance traveled. Write a function to show the relationship between number of miles traveled \begin{align*}(D)\end{align*} and number of hours \begin{align*}(t)\end{align*} traveled for each biker.
Solution:
We can use problem solving steps to help us to analyze the graph, fill in the table, and write a function.
\begin{align*}& \mathbf{Describe:} && \text{The graph is a Distance-Time graph and shows two lines. One line is} \\ & && \text{steeper than the other. One line starts where}\ t = 0.\ \text{The other line starts} \\ & && \text{where}\ t = 2.\ \text{The Fact identifies two bikers, Kelly and Finley}. \\ & && \text{Kelly left 2 hours after Finley and biked at a faster speed}. \\ \\ & \mathbf{My \ Job:} && \text{Use the graph and the Fact to figure out which line represents each biker}. \\ & && \text{Use the graph data to complete the table for each biker. Write the function} \\ & && \text{that shows the relationship between Distance and Time for each biker}. \\ \\ & \mathbf{Plan:} && \text{Use the Fact and graph to identify the line that represents Kelly} \\ & && \text{Complete the table for Kelly using points on the line. Write the} \\ & && \text{function rule. Do the same for Finley}. \\ \\ & \mathbf{Solve:} && \text{Since Kelly leaves later than Finley, the green line represents Kelly. Kelly's} \\ & && \text{number of hours is two less than the number of hours Finley bikes}. \\ & && \text{Kelly's line is steeper. This means that Kelly is biking faster than Finley}.\end{align*}
Time (Number of hours) | Kelly Distance (Number of miles) | Finley Distance (Number of miles) |
---|---|---|
0 | 0 | 0 |
1 | 0 | 6 |
2 | 0 | 12 |
3 | 8 | 18 |
4 | 16 | 24 |
5 | 24 | 30 |
\begin{align*}& && \text{Kelly}:\ D = 8(t - 2) \\ & && \text{Finley}:\ D = 6t\end{align*}
\begin{align*}& \mathbf{Check:} && \text{Replace variables in the functions with their values. Check results}\\ & &&\text{with data in the table}.\end{align*}
Kelly \begin{align*}D = 8(t - 2)\end{align*} | Finley \begin{align*}D = 6t\end{align*} | |
---|---|---|
For \begin{align*}t = 1\end{align*} | \begin{align*}6 = 6 \times 1\end{align*} | |
For \begin{align*}t = 2\end{align*} | \begin{align*}0 = 8(2 - 2)\end{align*} | \begin{align*}12 = 6 \times 2\end{align*} |
For \begin{align*}t = 3\end{align*} | \begin{align*}8 = 8(3 - 2)\end{align*} | \begin{align*}18 = 6 \times 3\end{align*} |
For \begin{align*}t = 4\end{align*} | \begin{align*}16 = 8(4 - 2)\end{align*} | \begin{align*}24 = 6 \times 4\end{align*} |
For \begin{align*}t = 5\end{align*} | \begin{align*}24 = 8(5 - 2)\end{align*} | \begin{align*}30 = 6 \times 5\end{align*} |
Example C
Fact: Robinson left 1 hour after Cranston and biked at half of Cranston’s speed.
Use the Fact and the graph. Complete the table for each biker showing distance traveled. Write a function to show the relationship between number of miles traveled \begin{align*}(D)\end{align*} and number of hours \begin{align*}(t)\end{align*} traveled for each biker.
Solution:
We can use problem solving steps to help us to analyze the graph, fill in the table, and write a function.
\begin{align*}& \mathbf{Describe:} && \text{The graph is a Distance-Time graph and shows two lines. One line is} \\ & && \text{steeper than the other. One line starts where}\ t = 0.\ \text{The other line starts} \\ & && \text{where}\ t = 1.\ \text{The Fact identifies two bikers, Robinson and Cranston}. \\ & && \text{Robinson left 1 hour after Cranston and biked at half of Cranston's speed}. \\ \\ & \mathbf{My \ Job:} && \text{Use the graph and the Fact to figure out which line represents each biker}. \\ & && \text{Use the graph data to complete the table for each biker. Write the function} \\ & && \text{that shows the relationship between Distance and Time for each biker}. \\ \\ & \mathbf{Plan:} && \text{Use the Fact and graph to identify the line that represents Robinson} \\ & && \text{Complete the table for Robinson using points on the line. Write the} \\ & && \text{function rule. Do the same for Cranston}. \\ \\ & \mathbf{Solve:} && \text{Since Robinson leaves later than Cranston, the green line represents Robinson. Robinson's} \\ & && \text{number of hours is one less than the number of hours Cranston bikes}. \\ & && \text{Cranston's line is steeper. This means that Cranston is biking faster than Connors}.\end{align*}
Time (Number of hours) | Robinson Distance (Number of miles) | Cranston Distance (Number of miles) |
---|---|---|
0 | 0 | 0 |
1 | 0 | 10 |
2 | 5 | 20 |
3 | 10 | 30 |
4 | 15 | 40 |
5 | 20 | 50 |
\begin{align*}& && \text{Robinson}:\ D = 5(t - 1) \\ & && \text{Cranston}:\ D = 10t\end{align*}
\begin{align*}& \mathbf{Check:} && \text{Replace variables in the functions with their values. Check results}\\ & &&\text{with data in the table}.\end{align*}
Robinson \begin{align*}D = 5(t - 1)\end{align*} | Cranston \begin{align*}D = 10t\end{align*} | |
---|---|---|
For \begin{align*}t = 1\end{align*} | \begin{align*}0 = 5(1 - 1)\end{align*} | \begin{align*}10 = 10 \times 1\end{align*} |
For \begin{align*}t = 2\end{align*} | \begin{align*}5 = 5(2 - 1)\end{align*} | \begin{align*}20 = 10 \times 2\end{align*} |
For \begin{align*}t = 3\end{align*} | \begin{align*}10 = 5(3 - 1)\end{align*} | \begin{align*}30 = 10 \times 3\end{align*} |
For \begin{align*}t = 4\end{align*} | \begin{align*}15 = 5(4 - 1)\end{align*} | \begin{align*}40 = 10 \times 4\end{align*} |
For \begin{align*}t = 5\end{align*} | \begin{align*}20 = 5(5 - 1)\end{align*} | \begin{align*}50 = 10 \times 5\end{align*} |
Concept Problem Revisited
Fact: Judson left on the bike hike one hour after Connor and biked at a faster speed than Connor.
We can use problem solving steps to help us to analyze the graph, fill in the table, and write a function.
\begin{align*}& \mathbf{Describe:} && \text{The graph is a Distance-Time graph and shows two lines. One line is} \\ & && \text{steeper than the other. One line starts where}\ t = 0.\ \text{The other line starts} \\ & && \text{where}\ t = 1.\ \text{The Fact identifies two bikers, Judson and Connors}. \\ & && \text{Judson left one hour after Connors and rode faster than Connors}. \\ \\ & \mathbf{My \ Job:} && \text{Use the graph and the Fact to figure out which line represents each biker}. \\ & && \text{Use the graph data to complete the table for each biker. Write the function} \\ & && \text{that shows the relationship between Distance and Time for each biker}. \\ \\ & \mathbf{Plan:} && \text{Use the Fact and graph to identify the line that represents Judson} \\ & && \text{Complete the table for Judson using points on the line. Write the} \\ & && \text{function rule. Do the same for Connors}. \\ \\ & \mathbf{Solve:} && \text{Since Judson leaves later than Connors,}\ \mathbf{line\ b}\ \text{represents Judson. Judson's} \\ & && \text{number of hours is one less than the number of hours Connors bikes}. \\ & && \text{Judson's line is steeper. This means that Judson is biking faster than Connors}.\end{align*}
Time (Number of hours) | Judson Distance (Number of miles) | Connors Distance (Number of miles) |
---|---|---|
0 | 0 | 0 |
1 | 0 | 5 |
2 | 10 | 10 |
3 | 20 | 15 |
4 | 30 | 20 |
5 | 40 | 25 |
\begin{align*}& && \text{Judson}:\ D = 10(t - 1) \\ & && \text{Connors}:\ D = 5t\end{align*}
\begin{align*}& \mathbf{Check:} && \text{Replace variables in the functions with their values. Check results}\\ & &&\text{with data in the table}.\end{align*}
Judson \begin{align*}D = 10(t - 1)\end{align*} | Connors \begin{align*}D = 5t\end{align*} | |
---|---|---|
For \begin{align*}t = 1\end{align*} | \begin{align*}0 = 10(1 - 1)\end{align*} | \begin{align*}5 = 5 \times 1\end{align*} |
For \begin{align*}t = 2\end{align*} | \begin{align*}10 = 10(2 - 1)\end{align*} | \begin{align*}10 = 5 \times 2\end{align*} |
For \begin{align*}t = 3\end{align*} | \begin{align*}20 = 10(3 - 1)\end{align*} | \begin{align*}15 = 5 \times 3\end{align*} |
For \begin{align*}t = 4\end{align*} | \begin{align*}30 = 10(4 - 1)\end{align*} | \begin{align*}20 = 5 \times 4\end{align*} |
For \begin{align*}t = 5\end{align*} | \begin{align*}40 = 10(5 - 1)\end{align*} | \begin{align*}25 = 5 \times 5\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 or function is an equation that can describe the relationship between the variables in a graph or a table. In this concept, we wrote functions that showed the relationship between the number of hours and the number of miles.
Guided Practice
1. Fact: Landis left 3 hours after Bricknell and biked twice as fast as Bricknell.
- Use the Fact and the graph. Complete the table for each biker showing distance traveled. Write a function to show the relationship between number of miles traveled \begin{align*}(D)\end{align*} and number of hours \begin{align*}(t)\end{align*} traveled for each biker.
2. Fact: Jefferson biked one-third Richards speed. Jefferson left 2 hours before Richards.
- Use the Fact and the graph. Complete the table for each biker showing distance traveled. Write a function to show the relationship between number of miles traveled \begin{align*}(D)\end{align*} and number of hours \begin{align*}(t)\end{align*} traveled for each biker.
3. Fact: Prentiss biked twice as fast as Jerome and left 2 hours after Jerome.
- Use the Fact and the graph. Complete the table for each biker showing distance traveled. Write a function to show the relationship between number of miles traveled \begin{align*}(D)\end{align*} and number of hours \begin{align*}(t)\end{align*} traveled for each biker.
Answers:
1. Landis: \begin{align*}D = 8(t-3)\end{align*}; Bricknell: \begin{align*}D = 4t\end{align*}
Time (Number of hours) | Landis Distance (Number of miles) | Bricknell Distance (Number of miles) |
---|---|---|
0 | 0 | 0 |
1 | 0 | 4 |
2 | 0 | 8 |
3 | 0 | 12 |
4 | 8 | 16 |
5 | 16 | 20 |
2. Jefferson: \begin{align*}D = 6t\end{align*}; Richards: \begin{align*}D = 18(t - 2)\end{align*}
Time (Number of hours) | Jefferson Distance (Number of miles) | Richards Distance (Number of miles) |
---|---|---|
0 | 0 | 0 |
1 | 6 | 0 |
2 | 12 | 0 |
3 | 18 | 18 |
4 | 24 | 36 |
5 | 30 | 54 |
3. Prentiss: \begin{align*}D = 16(t - 2)\end{align*}; Jerome: \begin{align*}D = 8t\end{align*}
Time (Number of hours) | Prentiss Distance (Number of miles) | Jerome Distance (Number of miles) |
---|---|---|
0 | 0 | 0 |
1 | 0 | 8 |
2 | 0 | 16 |
3 | 16 | 24 |
4 | 32 | 32 |
5 | 48 | 40 |
Practice
Fact: Evan biked three times as fast as Jake, but left 2 hours later.
- Use the Fact and the graph. Complete a table for each biker showing distance traveled.
- Write a function to show the relationship between number of miles traveled \begin{align*}(D)\end{align*} and number of hours \begin{align*}(t)\end{align*} traveled for each biker.
Fact: Josh left before Sam, but biked half as fast as Sam.
- Use the Fact and the graph. Complete a table for each biker showing distance traveled.
- Write a function to show the relationship between number of miles traveled \begin{align*}(D)\end{align*} and number of hours \begin{align*}(t)\end{align*} traveled for each biker.
Fact: Katie biked one third as fast as Beth, but left before her.
- Use the Fact and the graph. Complete a table for each biker showing distance traveled.
- Write a function to show the relationship between number of miles traveled \begin{align*}(D)\end{align*} and number of hours \begin{align*}(t)\end{align*} traveled for each biker.
Fact: Whitney left one hour after Jack.
- Use the Fact and the graph. Complete a table for each biker showing distance traveled.
- Write a function to show the relationship between number of miles traveled \begin{align*}(D)\end{align*} and number of hours \begin{align*}(t)\end{align*} traveled for each biker.
- Lindsey bikes twice as fast as Thomas, but leaves 2 hours after him. Thomas bikes 8 miles per hour. Create a graph that matches this situation.
- Amy bikes one third as fast as Mark, but leaves 3 hours before him. Mark bikes 15 miles per hour. Create a graph that matches this situation.
Image Attributions
Students analyze graphs to make tables and write functions to show the relationship between number of hours and distance traveled. Students use problem solving steps to help.