In this section we will explore two relationships between variables.
The first is Direct Variation. Direct variation, in more common terms, means proportional. As one value increases, the other will increase by the same factor. If y varies directly as x, then the y value will increase or decrease by the same constant. For example, if the x value is doubled the y value will also double, if the x value is cut by a third the y value will also be cut by a third.
The constant factor is known as the constant of variation (k). The image below shows the rule for direct variation: y = kx. Direct variation is a linear function, and will always go through the origin.
Notice in the table of values, y/x equals 2 with each coordinate. This is one of the ways the constant of variation can be identified. k = y/x.
The first is Direct Variation. Direct variation, in more common terms, means proportional. As one value increases, the other will increase by the same factor. If y varies directly as x, then the y value will increase or decrease by the same constant. For example, if the x value is doubled the y value will also double, if the x value is cut by a third the y value will also be cut by a third.
The constant factor is known as the constant of variation (k). The image below shows the rule for direct variation: y = kx. Direct variation is a linear function, and will always go through the origin.
Notice in the table of values, y/x equals 2 with each coordinate. This is one of the ways the constant of variation can be identified. k = y/x.
The second type of variation is Inverse Variation. When 2 values are inversely proportionate, one value will increase as the second value will decrease. If one value is multiplied by 3, the 2nd value will be divided by 3. If one number is divided by 2, the other will be multiplied by 2.
This is a rational function and will graph a hyperbola. The rule is y = k/x. The vertical asymptote is x = 0 since zero cannot be in the denominator.
Inverse variation causes the product of the values to be constant, so the constant of variation can be found by multiplying the two values together. Notice in the table of values below, x times y always equals 2. k = xy
This is a rational function and will graph a hyperbola. The rule is y = k/x. The vertical asymptote is x = 0 since zero cannot be in the denominator.
Inverse variation causes the product of the values to be constant, so the constant of variation can be found by multiplying the two values together. Notice in the table of values below, x times y always equals 2. k = xy
Examples
As you saw in the example videos, there are other combinations of direct and inverse variations you will need to know. Joint variation and Combined variations are shown in the diagram below. The problems are worked out in the same manor as simple direct and inverse variation problems, but usually involve an extra variable.
These types of variations can include squares and roots. Watch the example video below and view the examples.
Example: Variations with Powers and Roots | |
File Size: | 6030 kb |
File Type: | mp4 |
Complete the following Worksheet and check your answers. The problems follow the examples from above in case you need to go back and review.
When you have finished, click the practice link to complete the online skill.
When you have finished, click the practice link to complete the online skill.