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 and (for positive integer n) vary directly then:
- It is denoted as
- for some constant k
- This can be written as
- 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 and
- It could be and
- Use the given information to find their constant ratio k
- Also called constant of proportionality
- Substitute the given values of and into your formula
- Solve to find k
- Write the equation which models their relationship
- 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, milliseconds, varies directly with the square of the number of items, , in the list. The computer program takes 48 milliseconds to order a list with 8 items.
a)
Find an equation connecting and .
b)
Find the time it takes to order a list of 50 numbers.
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 and (for positive integer n) vary inversely then:
- It is denoted
- for some constant k
- This can be written
- The graphs of these models all have a vertical asymptote at the y-axis
- This means that as gets closer to 0 the absolute value of gets further away from 0
- can never equal 0
How do I solve inverse variation problems?
- Identify which two variables vary inversely
- It might not be and
- It could be and
- Use the given information to find their constant product k
- Also called constant of proportionality
- Substitute the given values of and into your formula
- Solve to find k
- Write the equation which models their relationship
- You can then use the equation to solve problems
Exam 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, hours, it takes to complete a project varies inversely to the number of people working on it, . If 4 people work on the project it takes 70 hours to complete.
a)
Write an equation connecting and .
b)
Given that the project needs to be completed within 18 hours, find the minimum number of people needed to work on it.