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

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Dan

Last updated

20 June 2023

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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 $y$ and $x^{n}$ (for positive integer n) vary directly then:
- It is denoted as $y \propto x^{n}$
- $y = k x^{n}$ for some constant k
  - This can be written as $\frac{y}{x^{n}} = k$
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 $x^{3}$ 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
- $y = k x^{n}$
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.

Find an equation connecting

t

and

n

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

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 $x^{n}$ (for positive integer n) vary inversely then:
- It is denoted $y \propto \frac{1}{x^{n}}$
- $y = \frac{k}{x^{n}}$ for some constant k
  - This can be written $x^{n} y = k$
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 $x^{3}$ 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
- $y = \frac{k}{x^{n}}$
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.

Write an equation connecting

t

and

n

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

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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