# 7.6 – Inverse Variation

## Key Terms

• Direct Variation – A relationship in which one variable is a constant multiple of the other variable.
• You can say that y varies directly with x and write y = kx, where k is a constant.
• $Output=\;Constant\bullet Input$
• $Height=\;Constant\bullet Width$
• $\frac{Height}{Width}=Constant$
• Input – A number that is entered into a function.
• The input variable in a function is the independent variable.
• Inverse Variation – A relationship in which quantities change inversely — as one quantity gets bigger, the other one gets smaller.
• $Output=\frac{Constant}{Input}$
• $Height=\frac{Constant}{Width}$
• $Height\;\bullet Width=Constant$
• Output – The result of a function.
• The output variable in a function is the dependent variable.

## Notes

Direct Variation Inverse Variation
• A proportional relationship between two variables whose ratio is a constant.
• $Output=Constant\;\bullet Input$
• To find the constant: $constant=\frac{y}{x}$
• Input and output change in the same way proportionally.
• As input increases, output increases.
• As input decreases, output decreases.
• If you double the input, the output doubles.
• The size changes, but the shape does not.
• The equation is linear, so the graph is a line.

• A proportional relationship between two variables whose product is a constant.
• $Output=\frac{Constant}{Input}$
• To find the constant: $constant=y\bullet x$
• Input and output change in opposite ways proportionally.
• As input increases, output decreases.
• As input decreases, output increases.
• If you double the input, the output is halved.
• The shape changes, but the size does not.
• The equation is rational, so the graph is a curve.

Direct Variation Vs Inverse Variation
• Animation: Direct Variation

• Animation: Inverse Variation

## Examples

 Ex 1. Find the constant. Direct variation: width = 10, height = 5 Width is the input and height is the output. $Output=Constant\;\bullet Input$, so $height=Constant\;\bullet width$ $5=c\bullet 10$ Divide both sides by 10: $\frac{5}{10}=\frac{10c}{10}$ Answer: $c=\frac{1}{2}$ Ex 2. Ex 2. Find the constant. Inverse variation: width = 10, height = 11 Width is the input and height is the output. $Output=\frac{Constant}{Input}$, so $10=\frac{Constant}{10}$ Multiply 10 on both sides: $11\bullet 10=\frac{c\bullet 10}{10}$ Answer: $c=110$