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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.

Alg2B 7.5 Direct Variation

  • 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.

Alg2B 7.5 Inverse Variation

Direct Variation Vs Inverse Variation
Alg2B 7.5 Compare
  • 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

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