PR - Direct Variation Lesson

Direct Variation

In this lesson we will learn how to represent proportional relationships using the form y = kx. Basically, this equation represents the relationship between x and y, where one is a constant multiple of the other. In other words, when one variable changes, the other variable changes by a constant factor (k). This constant is known as the constant of proportionality.

There are several ways to represent proportional relationships. They can be represented by a table, a graph, and an equation. You can read direct variation as " y varies directly as x" or "y is directly proportional to x" or "y varies with x."

Example 1 - Equations

We will begin with direct variation equations. Look at the equations below and determine whether each represents a direct variation. If so, we need to identify the constant of proportionality. (Remember, the direct variation equation is y = kx)

1.

$y+4=x\:\text{Solve this equation for y}\\ \:\:-4\:\:\:-4\\ y=x-4\:\text{(This is not in the form y=kx, so it is not direct variation.)}$

2.

$3y=2x\:\text{Solve this equation for y}\\ \frac{3y}{3}=\frac{2x}{3}\\ y=\frac{2}{3}x$

This is in the form y = kx and the constant of proportionality is $\frac{2}{3}$.

There is another way to quickly see if an equation is direct variation. Plug in 0 for both the x and y values. We are going to try this for the same equations and see what happens. This strategy does not help you find the constant and can only be used for determination.

$y+4=x\:\text{(Plug in 0 for y and x)}\\ 0+4=0\:\text{(This is false, so the equation is not direct variation.)}\\ 3y=2x\:\text{(Plug in 0 for y and x.)}\\ 3(0)=2(0)\\ 0=0\:\text{(Direct Variation)}$

Example 2 - Function Table

Look at the function tables below. Can you see the pattern? Remember, when you have a table giving you the x and y values, you simply divide y by x to see if you get a constant (K). If you do not get the same value, then the function table does not represent direct variation.

y = kx
x	y
0	0
1	8
2	16
3	24
4	32
5	40

 

$y\ne kx$
x	1	2	3	4	5	6	7	8
y	4	5	6	7	8	9	10	11

Tables can be used to write direct variation equations. When you divide y by x and get the same value, it will become k. For example, in the first direct variation table above, each time you divide y by x, you should get 8. This means that 8 is the constant and that each time x increases by 1, y increases by 8. Once you have this information, you can write an equation. The equation for this table would be: $y=8x$

Example 3 - Graphs

The graph of a direct variation (proportional relationship) is a straight line that passes through the origin. On a direct variation graph, the graph will always pass through (1, k), where k is the constant of proportionality. When x is 1, this also represents the unit rate.

Take a look at the graphs below and note differences in the graph that represents direct variation and the graph that does not represent direct variation.

Notice that in the first graph the line does not pass through (0, 0). This is not a direct variation.

In the second graph, the graph shows a straight line through (0, 0), so it does represent direct variation (proportional relationship). The Constant of proportionality is $\frac{1}{2}$, so $k=-\frac{1}{2}$. Notice that the line goes downward from left to right - which means negative slope. If the constant is positive, the line will slant upward from left to right. An equation can be written from this graph once you find the constant of proportionality. The equation is $y=-\frac{1}{2}x$.

Equations, tables, and graphs may be used to represent proportional relationships or direct variation. You can use any of these representations interchangeably. You can use any of these depictions to solve word problems involving direct variation.

A bicycle travels at a speed of 10 miles per hour. Write a direct variation equation for the distance, y, the bike travels in x hours.   (The constant of proportionality is 10 because each time the bicycle travels 1 hour; it will go another 10 miles.)

y = 10x 

A bicycle travels at a speed of 10 miles per hour. Graph the data.

First, make a table. Since our subject is speed, negative numbers will not be used.

y = kx
$x_{hours}$	1	2	3	4	5	6	7	8
$y_{miles}$	10	20	30	40	50	60	70	80

Use the ordered pairs to plot the points on a coordinate plane. Connect the points in a straight line. Label each axis.

Direct Variation Practice

Determine if the table, graph, equation, or word problem represents direct variation (proportional relationship). If yes, identify the constant of proportionality (k). Rollover each to check your work!

Problem: 5y = 80x

Solution: Yes; k=16

Problem: 2y = 18x + 12

Solution: No; there is no constant proportionality (k)

Problem: 3y = 37.2x

Solution: Yes; k = 12.4

Solution: Yes; k = 7.25

Solution: Yes; k = 15

Solution: No; there is no constant of proportionality

x	0	1	3	5
y	0	4	12	20

Solution: Yes; k = 4

x	3	9	18
y	1	4	7

Solution: No; there is no constant of proportionality

A pipeline delivers 12,000 gallons of natural gas every other month.

Solution: Yes; k = 6000

An empty swimming pool is being filled at a rate of 5 gallons per minute.

Solution: Yes; k= 5

Direct Variation Homework

Now that you have spent some time learning about direct variation, you are ready to complete your Ratio and Proportional Relationships: Direct Variation Homework. Download your homework by CLICKING HERE Links to an external site..

Once you have completed your homework, AND MAKE SURE YOU ATTEMPTED AND WORKED THE PROBLEMS OUT ON YOUR OWN, click here to download your homework key. Links to an external site.

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