3.06 Interpret graphs of proportional relationships
Introduction
We've learned that proportional relationships are a special kind of linear relationship that can be written generally in the form y = kx . The x and y quantities vary in such a way that there is a constant multiplier between them. In other words, they always vary by the same constant.
Interpret proportional relationships
To interpret information from a graph, we need to look at pairs of coordinates. Coordinates tell us how one variable relates to the other. Each pair has an x-value and a y-value in the form (x, y).
- The x-value tells us the value of the variable on the horizontal axis.
- The y-value tells us the value of the variable on the vertical axis.
It doesn't matter what labels we give our axes, this order is always the same.
One point that will always appear in a proportional relationship is \left(0,0\right). This means that every proportional relationship will pass through the origin. This is because every proportional relationship can be represented with the equation y=kx:
| \displaystyle y | \displaystyle = | \displaystyle kx | Given equation |
| \displaystyle y | \displaystyle = | \displaystyle k\times0 | Let x=0 |
| \displaystyle y | \displaystyle = | \displaystyle 0 | Simplify |
Another point in a proportional relationship that is interesting to consider is when x=1, which looks like \left(1,y\right).
| \displaystyle y | \displaystyle = | \displaystyle kx | Given equation |
| \displaystyle y | \displaystyle = | \displaystyle k\times1 | Let x=1 |
| \displaystyle y | \displaystyle = | \displaystyle k | Simplify |
This means that when x=1, y is equal to k, the constant of proportionality. The coordinate will always be in the form \left(1,k\right).
Let's look at an example of a graph of a proportional relationship that represents the distance traveled over time.
As you can see, there are a few points labeled. From left to right, the first point is \left(1,2\right). This means that for 2 miles traveled, it takes 1 minute.
The next point is \left(2,4\right). This means that for 4 miles traveled, it takes 2 minutes.
The last point is \left(4,8\right). This means that for 8 miles traveled, it takes 4 minutes.
We can see that this is a change of a constant rate of \dfrac{2 \text{ miles}}{1 \text{ minute}}.
Examples
Example 1
The number of liters of gas used by a fighter jet over a certain number of seconds is shown in the graph:
What does the point on the graph represent?
Proportional relationships are a special kind of linear relationship that can be written generally in the form y = kx.
To interpret information from a graph, we need to look at pairs of coordinates that have an x-value and a y-value in the form (x, y).
- The x-value tells us the value of the variable on the horizontal axis.
- The y-value tells us the value of the variable on the vertical axis.
Proportional relationships will always pass through the origin, \left(0,0\right).
When x=1, the coordinate will be \left(1,k\right).