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Proportional Functions

Two variables x and y are proportional if the ratio between them is constant. That means you always get the same answer when you divide y by x, regardless of where you are along the two graphs.


Proportional Variables

Two variables, x and y, are proportional if

y = kx

where k is a constant.

Example 1

Saying that y = kx is the same as saying that y x = k: y x = k x y x = k x y = kx

Proportional functions are actually just a special case of linear functions, f(x) = ax + b. What has happened is that b = 0, so b is gone. This means that the graph intersects the y-axis at the origin each time. In addition, the slope a is replaced by the proportionality constant k. Here are a few examples:

Example 2

This graph shows y = x, that is, k = 1. Since the graph is proportional, for all the coordinates on the graph, if you divide the y-coordinate by the x-coordinate, the answer is k = 1.

The graph of y=x

Example 3

Is the graph y = 2x 3 proportional?

You can find that out by doing a few conversions: y = 2x 3 = 2 × x 3 × 1 = 2 3 ×x 1 0.67 × x = 0.67x

You have determined that k 0.67 or k = 2 3, so the graph is proportional.

Example 4

You have been given the following points:

x-values 1 2 3 4 5

y -values 7 14 21 28 35

Are these points part of a proportional function?

From the theory, you know that if you divide the y-value by the x-value, and the answer is the same each time, all points belong to a proportional function. You check the points from the table: 7 1 = 7 14 2 = 7 21 3 = 7 28 4 = 7 35 5 = 7

Because all the answers are the same, you are working with a proportional function. The function is f(x) = 7x.

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