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IBMaths: AI HLDPRevision Notes2. Functions2.3 Modelling with Functions2.3.4 Direct & Inverse Variation

Direct & Inverse Variation (DP IB Maths: AI HL)

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Dan

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Dan

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Direct Variation

What is direct variation?

  • Two variables are said to vary directly if their ratio is constant (k)
    • This is also called direct proportion
  • If yand x to the power of n (for positive integer n) vary directly then:
    • It is denoted as y proportional to x to the power of n
    • y equals k x to the power of n for some constant k
      • This can be written as y over x to the power of n equals k
  • The graphs of these models always start at the origin

How do I solve direct variation problems?

  • Identify which two variables vary directly
    • It might not be x and y
    • It could be x cubed and y
  • Use the given information to find their constant ratio k
    • Also called constant of proportionality
    • Substitute the given values of x and y into your formula
    • Solve to find k
  • Write the equation which models their relationship
    • y equals k x to the power of n
  • You can then use the equation to solve problems

Worked example

A computer program sorts a list of numbers into ascending order. The time it takes, t milliseconds, varies directly with the square of the number of items, n, in the list. The computer program takes 48 milliseconds to order a list with 8 items.

a)
Find an equation connecting t and n.

2-3-4-ib-ai-sl-direct-variation-a-we-solution

b)
Find the time it takes to order a list of 50 numbers.

2-3-4-ib-ai-sl-direct-variation-b-we-solution

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Inverse Variation

What is inverse variation?

  • Two variables are said to vary inversely if their product is constant (k)
    • This is also called inverse proportion
  • If y and x to the power of n (for positive integer n) vary inversely then:
    • It is denoted y proportional to 1 over x to the power of n
    • y equals k over x to the power of n for some constant k
      • This can be written x to the power of n y equals k
  • The graphs of these models all have a vertical asymptote at the y-axis
    • This means that as x gets closer to 0 the absolute value of y gets further away from 0
    • x can never equal 0

How do I solve inverse variation problems?

  • Identify which two variables vary inversely
    • It might not be x and y
    • It could be x cubed and y
  • Use the given information to find their constant product k
    • Also called constant of proportionality
    • Substitute the given values of x and y into your formula
    • Solve to find k
  • Write the equation which models their relationship
    • y equals k over x to the power of n
  • You can then use the equation to solve problems

Examiner Tip

  • Reciprocal graphs generally have two parts/curves
    • Only one – usually the positive – may be relevant to the model
    • Think about why x/t/θ can only take positive values - refer to the context of the question

Worked example

The time, t hours, it takes to complete a project varies inversely to the number of people working on it, n. If 4 people work on the project it takes 70 hours to complete.

a)
Write an equation connecting t and n.

2-3-4-ib-ai-sl-inverse-variation-a-we-solution

b)
Given that the project needs to be completed within 18 hours, find the minimum number of people needed to work on it.

2-3-4-ib-ai-sl-inverse-variation-b-we-solution

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    Dan

    Author: Dan

    Expertise: Maths

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.