3.1.2: Graphs of Proportional Relationships

Lesson

Let's think about scale.

Exercise \(\PageIndex{1}\): An Unknown Situation

Here is a graph that could represent a variety of different situations.

Figure \(\PageIndex{1}\)

1. Write an equation for the graph.
2. Sketch a new graph of this relationship.

Figure \(\PageIndex{2}\)

Exercise \(\PageIndex{2}\): Card Sort: Proportional Relationships

Your teacher will give you 12 graphs of proportional relationships.

1. Sort the graphs into groups based on what proportional relationship they represent.
2. Write an equation for each different proportional relationship you find.

Exercise \(\PageIndex{3}\): Different Scales

Two large water tanks are filling with water. Tank A is not filled at a constant rate, and the relationship between its volume of water and time is graphed on each set of axes. Tank B is filled at a constant rate of \(\frac{1}{2}\) liters per minute. The relationship between its volume of water and time can be described by the equation \(v=\frac{1}{2}t\), where \(t\) is the time in minutes and \(v\) is the total volume in liters of water in the tank.

Figure \(\PageIndex{3}\): Graph, horizontal axis, time in minutes, scale 0 to 1 and 8 tenths, by 2 tenth's. vertical axis, volume in liters, 0 to 1 and 8 tenths, by 2 tenth's. curved line passing through origin, 1 comma 1 and 1 and 4 tenths comma 1 and 3 tenths.

Figure \(\PageIndex{4}\): Graph, horizontal axis, time in minutes, scale 0 to 80, by 20's. vertical axis, volume in liters, 0 to 50, by 10's. curved line passing through origin, 20 comma 12 and 60 comma 30.

1. Sketch and label a graph of the relationship between the volume of water \(v\) and time \(t\) for Tank B on each of the axes.
2. Answer the following questions and say which graph you used to find your answer.
   1. After 30 seconds, which tank has the most water?
   2. At approximately what times do both tanks have the same amount of water?
   3. At approximately what times do both tanks contain 1 liter of water? 20 liters?

Are you ready for more?

A giant tortoise travels at 0.17 miles per hour and an arctic hare travels at 37 miles per hour.

1. Draw separate graphs that show the relationship between time elapsed, in hours, and distance traveled, in miles, for both the tortoise and the hare.
2. Would it be helpful to try to put both graphs on the same pair of axes? Why or why not?
3. The tortoise and the hare start out together and after half an hour the hare stops to take a rest. How long does it take the tortoise to catch up?

Summary

The scales we choose when graphing a relationship often depend on what information we want to know. For example, say two water tanks are filled at different constant rates. The relationship between time in minutes \(t\) and volume in liters \(v\) of tank A is given by \(v=2.2t\).

For tank B the relationship is \(v=2.75t\).

These equations tell us that tank A is being filled at a constant rate of 2.2 liters per minute and tank B is being filled at a constant rate of 2.75 liters per minute.

If we want to use graphs to see at what times the two tanks will have 110 liters of water, then using an axis scale from 0 to 10, as shown here, isn't very helpful.

Figure \(\PageIndex{5}\): Graph, horizontal axis, time in minutes, scale 0 to 9, by 1's. vertical axis, volume in liters, 0 to 9, by 1's. lines, first line passing through origin and 2 comma 5 and 5 tenths. second line passing through origin and 2 comma 4 and 5 tenths.

If we use a vertical scale that goes to 150 liters, a bit beyond the 110 we are looking for, and a horizontal scale that goes to 100 minutes, we get a much more useful set of axes for answering our question.

Now we can see that the two tanks will reach 110 liters 10 minutes apart—tank B after 40 minutes of filling and tank A after 50 minutes of filling.

It is important to note that both of these graphs are correct, but one uses a range of values that helps answer the question. In order to always pick a helpful scale, we should consider the situation and the questions asked about it.

Figure \(\PageIndex{6}\): Graph, horizontal axis, time in minutes, scale 0 to 90, by 10's. vertical axis, volume in liters, 0 to 140, by 10's. 2 lines, first line passing through origin and 40 comma 110. second line passing through origin and 50 comma 110.

Glossary Entries

Definition: Constant of Proportionality

In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity. This number is called the constant of proportionality.

In this example, the constant of proportionality is 3, because \(2\cdot 3=6\), \(3\cdot 3=9\), and \(5\cdot 3=15\). This means that there are 3 apples for every 1 orange in the fruit salad.

Table \(\PageIndex{1}\)

number of oranges	number of apples
\(2\)	\(6\)
\(3\)	\(9\)
\(5\)	\(15\)

Practice

Exercise \(\PageIndex{4}\)

The tortoise and the hare are having a race. After the hare runs 16 miles the tortoise has only run 4 miles.

The relationship between the distance \(x\) the tortoise “runs” in miles for every \(y\) miles the hare runs is \(y=4x\). Graph this relationship.

Figure \(\PageIndex{7}\)

Exercise \(\PageIndex{5}\)

The table shows a proportional relationship between the weight on a spring scale and the distance the spring has stretched.

1. Complete the table.
2. Describe the scales you could use on the \(x\) and \(y\) axes of a coordinate grid that would show all the distances and weights in the table.

Table \(\PageIndex{2}\)

distance (cm)	weight (newtons)
\(20\)	\(28\)
\(55\)
	\(140\)
\(1\)

Exercise \(\PageIndex{6}\)

Find a sequence of rotations, reflections, translations, and dilations showing that one figure is similar to the other. Be specific: give the amount and direction of a translation, a line of reflection, the center and angle of a rotation, and the center and scale factor of a dilation.

Figure \(\PageIndex{8}\): polar coordinate plane with center at A. quadrilateral BCDE and quadrilateral B prime prime, C prime prime, D prime prime, E prime prime graphed. B at 9 comma pi over 12. C at 6 comma pi over 4. D at 3 comma 5 pi over 12, E at 6 comma 11 pi over 6. B prime prime at 3 comma 17 pi over 12. C prime prime at 2 comma 5 pi over 4. D prime prime at 1 comma 13 pi over 12. E prime prime at 2 comma 5 pi over 3.

(From Unit 2.2.1)

Exercise \(\PageIndex{7}\)

Andre said, “I found two figures that are congruent, so they can’t be similar.”

Diego said, “No, they are similar! The scale factor is 1.”

Do you agree with either of them? Use the definition of similarity to explain your answer.

(From Unit 2.2.1)