# 8.2: Double Line Graphs

Difficulty Level: At Grade Created by: CK-12

Remember a line graph, by definition, can be the result of a linear function or can simply be a graph of plotted points, where the points are joined together by line segments. Line graphs that are linear functions are normally in the form \begin{align*}y = mx + b\end{align*}, where \begin{align*}m\end{align*} is the slope and \begin{align*}b\end{align*} is the \begin{align*}y\end{align*}-intercept. The graph below is an example of a linear equation with a slope of \begin{align*}\frac{2}{3}\end{align*} and a \begin{align*}y\end{align*}-intercept of \begin{align*}-2\end{align*}:

The second type of line graph is known as a broken-line graph. In a broken-line graph, the slope represents the rate of change, and the \begin{align*}y\end{align*}-intercept is actually the starting point. The graph below is a broken-line graph:

When the measurements began, the number of sales (the \begin{align*}y\end{align*}-intercept) was 645. The graph shows a significant increase in the number of sales from weeks 5 through 10 and a significant reduction in the number of sales from weeks 16 through 20.

In this lesson, you will be learning about comparing 2 line graphs that each contain data points. In statistics, when line graphs are in the form of broken-line graphs, they are of more use. Linear functions (i.e., \begin{align*}y = mx + b\end{align*}) are more for algebraic reasoning. Double line graphs, as with any double graphs, are often called parallel graphs, due to the fact that they allow for the quick comparison of 2 sets of data. In this chapter, you will see them referred to only as double graphs.

Example 1

Christopher and Jack are each opening businesses in their neighborhoods for the summer. Christopher is going to sell lemonade for \begin{align*}50 \cancel c\end{align*} per glass. Jack is going to sell popsicles for \$1.00 each. The following graph represents the sales for each boy for the 8 weeks in the summer.

a. Explain the slopes of the line segments for Christopher’s graph.

b. Explain the slopes of the line segments for Jack’s graph.

c. Are there any negative slopes? What does this mean?

d. Where is the highest point on Christopher’s graph? What does this tell you?

e. Where is the highest point on Jack’s graph? What does this tell you?

f. Can you provide some reasons for the shape of Jack’s graph?

g. Can you provide some reasons for the shape of Christopher’s graph?

Solution:

a. The slope of the line segments for Christopher’s graph (red) are positive for the first 5 weeks, meaning he was increasing his sales each week. This is also true from weeks 6 to 7. From weeks 5 to 6 and weeks 7 to 8, the slopes were decreasing, meaning there was a decrease in sales.

b. The same trend that is seen for Christopher’s graph (red) is also seen for Jack’s graph (blue). The slope of the line segments for Jack’s graph (blue) are positive for the first 5 weeks, meaning he was increasing his sales each week. This is also true from weeks 6 to 7. From weeks 5 to 6 and weeks 7 to 8, the slopes were decreasing, meaning there was a decrease in sales.

c. Negative sales from weeks 5 to 6 and weeks 7 to 8 (for both boys) mean there was a decrease in sales during these 2-week periods.

d. The highest point on Christopher’s graph occurred in week 7, when he sold 65 glasses of lemonade. This must have been a very good week\begin{align*}-\end{align*}nice and hot!

e. The highest point on Jack’s graph occurred in week 5, when he sold 74 popsicles. This must have been a very hot week as well!

f. Popsicles are a great food when you are warm and want a light snack. You can see how as the summer became hotter, the sales increased. Even in the weeks where it looks like Jack had a decrease in sales (maybe a few rainy days occurred, or it was not as hot), his sales still remained at a good level.

g. Lemonade is a very refreshing drink when you are warm. You can see how as the summer became hotter, the sales increased. Even in the weeks where it looks like Christopher had a decrease in sales (maybe a few rainy days occurred, or it was not as hot), his sales still remained at a good level, just as Jack's sales did.

Example 2

Thomas and Abby are training for the cross country meet at their school. Both students are in the 100 yard dash. The coach asks them to race 500 yards and time each 100 yard interval. The following graph represents the times for both Thomas and Abby for each of the five 100 yard intervals.

a. Who won the race? How do you know?

b. Between what times did Thomas (blue) appear to slow down? How do you know?

c. Between what times was Abby (red) ahead of Thomas? How do you know?

d. At what time did Thomas pass Abby? How do you know?

Solution:

a. Thomas (blue) won the race, because he finished the 500 yards in the least amount of time.

b. Between 20 and 40 seconds, Thomas (blue) seems to slow down, because the slope of the graph is less steep.

c. Between 0 and 57 seconds, Abby (pink) is ahead of Thomas (blue). You can see this, because the pink line is above the blue line.

d. At 57 seconds, Thomas (blue) passes Abby (pink). From this point onward, the blue line is above the pink line, meaning Thomas is running faster 100 yard intervals.

Example 3

Brenda and Ervin are each planting corn in a section of garden in their back yard. Brenda says that they need to put fertilizer on the plants 3 to 5 times per week. Ervin contradicts Brenda, saying that they need to fertilize only 1 to 2 times per week. Each gardener plants his or her garden of corn and measures the heights of their plants. The graph for the growth of their corn is found below:

a. Who was right? How do you know?

b. Between what times did Brenda’s (pink) garden appear to grow more? How do you know?

c. Between what times were Ervin's (blue) heights ahead of Brenda’s? How do you know?

Solution:

a. Brenda (pink) is correct, because her plants grew more in the same amount of time.

b. Between 4 and 8 weeks, Brenda’s plants seemed to grow faster (taller) than Ervin’s plants. You can tell this, because the pink line is above the blue line after the 4-week mark.

c. From 0 and 4 weeks, Ervin’s plants seemed to grow faster (taller) than Brenda’s plants. You can tell this, because the blue line is above the pink line before the 4-week mark.

Example 4

Nicholas and Jordan went on holidays with their families. They decided to monitor the mileage they traveled by keeping track of the time and the distance they were on the road. The boys collected the following data:

\begin{align*}&\text{Nicholas}\\ \\ & \text{Time (hr)} \qquad \qquad \quad 1 \qquad 2 \qquad 3 \qquad 4 \qquad 5 \qquad 6\\ & \text{Distance (miles)} \quad \quad 60 \quad 110 \quad 175 \quad 235 \quad 280 \quad 320\\ \\ &\text{Jordan}\\ \\ & \text{Time (hr)} \qquad \qquad \quad 1 \quad \ 2 \qquad 3 \qquad 4 \qquad 5 \qquad 6 \\ & \text{Distance (miles)} \qquad 50 \quad 90 \quad 125 \quad 125 \quad 165 \quad 210\end{align*}

a. Draw a graph to show the trip for each boy.

b. What conclusions could you draw by looking at the graphs?

Solution:

a.

You can also use TI technology to graph this data. First, you need to enter in all of the data Nicholas and Jordan collected.

Now you need to graph the 2 sets of data.

The resulting graph looks like the following:

b. Looking at the speed of Nicholas’s family vehicle and the shape of the graph, it could be concluded that Nicholas’s family was traveling on the highway going toward their family vacation destination. The family did not stop and continued on at a pretty steady speed until they reached where they were going.

Jordan’s trip was more relaxed. The speed indicates they were probably not on a highway, but more on country-type roads, and that they were traveling through a scenic route. In fact, from hours 3 to 4, the family stopped for some reason (maybe lunch), and then they continued on their way.

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