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Here are two graphs that could represent a variety of different situations.
Andre claims that the line in the graph on the left has a greater slope because it is steeper. Do you agree with Andre? Explain your reasoning.
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 liters per minute. The relationship between its volume of water and time can be described by the equation , where is the time in minutes, and is the total volume in liters of water in the tank.
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.
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.</p>
Answer the following questions and say which graph you used to find your answer.
Here are two ways to represent a situation.
Description:
The Origami Club is doing a car wash fundraiser to raise money for a trip. They charge the same price for every car. After 11 cars, they raised a total of \$93.50. After 23 cars, they raised a total of \$195.50.
| number of cars |
amount raised in dollars |
|---|---|
| 11 | 93.50 |
| 23 | 195.50 |
Create a graph that represents this situation.
The scales we choose when graphing a relationship often depend on what information we want to know. For example, consider two water tanks filled at different constant rates.
The relationship between time in minutes and volume in liters of Tank A can be described by the equation .
For Tank B the relationship can be described by the equation
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.
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.
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.
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.
What representation we choose for a proportional relationship also depends on our purpose. For example, if Tank C fills at a constant rate of 2.5 liters per minute and we want to see the change in volume every 30 minutes, we could use a table:
| minutes (t) | liters (v) |
|---|---|
| 0 | 0 |
| 30 | 75 |
| 60 | 150 |
| 90 | 225 |
No matter the representation or the scale used, the constant of proportionality is evident in each. In the equation, it is the number we multiply by. In the graph it is the slope, and in the table it is the number by which we multiply values in the left column to get values in the right column. We can think of the constant of proportionality as a rate of change: the amount one variable changes by when the other variable increases by 1. For Tank A, the rate of change of with respect to is 2.2 liters per minute.
The rate of change is the amount changes when increases by 1. On a graph, the rate of change is the slope of the line.
In this graph, increases by 15 dollars when increases by 1 hour. The rate of change is 15 dollars per hour.