Activity
Use the graphing tool or technology outside the course. Plot some values for the length and area of the garden with perimeter 50 meters on the coordinate plane.
Use your table, or here is a sample table of values if you need one:
| Length (meters) | Area (square meters) |
| 5 | 100 |
| 10 | 150 |
| 12 | 156 |
| 12.5 | 156.25 |
| 18 | 126 |
| 20 | 100 |
| 24 | 24 |
Compare your answer:
What do you notice about how the plotted points increase and decrease?
Compare your answer:
At first, the area grows as the length increases, but then it decreases. Also, the area increases more and more slowly as the length increases.
The points and each represent the length and area of the garden. Plot these two points on a coordinate plane, if you haven’t already done so. What do these points mean in this situation?
Compare your answer:
The point means that if the length of the garden is 3 meters, then the area is 66 square meters. This is right because the width is 22 meters and . The point means that the length of the garden is 22 meters, so its width is 3 meters, and the area is again 66 square meters.
Could the point represent the length and area of the garden?
Compare your answer:
The point cannot represent the length and area of the garden. If the length of the garden is 1 meter, the width would be 24 meters, so the area is 24 square meters, not 25 square meters.
Video: The Meaning of Points on a Graph
Watch the following video to learn more about the meaning of points on a graph.
Are you ready for more?
Extending Your Thinking
What happens to the area when you interchange the length and width? For example, compare the areas of a rectangle of length 11 meters and width 14 meters with a rectangle of length 14 meters and width 11 meters.
Compare your answer:
The area stays the same if the length and width are interchanged because the area is the product of the length and the width.
What patterns would you notice if you were to plot more length and area pairs on the graph?
Compare your answer:
Many of the areas would come with two different pairs: for example, 100 comes from the pair and also from the pair because a rectangle with length 5 meters and width 20 meters is the same as a rectangle with length 20 meters and width 5 meters. Using a different length and corresponding width, the 156 comes from the pair and also from the pair . The points on the graph would have reflectional symmetry about the line .
Self Check
The graph below shows a rectangle’s side length and its area. What is true about the point ?
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The area of the rectangle is 3 square meters.
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It is not possible to find the width of the rectangle when the length is 3 meters.
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The length of the rectangle is 3 meters, and the area is 21 square meters.
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The width of the rectangle is 3 meters, and the area is 21 square meters.
Additional Resources
Determining the Meaning of a Point on a Graph
The graph below shows the relationship between a rectangle’s side length and area.
What does the point mean in the situation?
The -axis represents the length of the rectangle in inches. So, when , the rectangle has a side length of 7 inches.
The -axis represents the area of the rectangle. So, when , the rectangle has an area of 28 square inches.
This also means the rectangle has a width of 4 inches since .
Try it
Try It: Determining the Meaning of a Point on a Graph
Using the same graph above, determine the meaning of the point and the width of the rectangle.
Here is how to determine the meaning of the point and the width of the rectangle:
Since the -axis is the length of the rectangle in inches, represents the length of the rectangle is 3 inches. Since the -axis represents the area in square inches, the area of the rectangle is 24 square inches.
This means the width of the rectangle must be 8 inches since .