Skip to main content
MathidiaMathematics Academy
Proportion and rates of change
Lesson 41 of 75
Introduction

Direct proportion with powers

Objective. Use y=kxⁿ models and determine constants.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Direct proportion with powers means y=kxⁿ, so y grows in proportion to a power of x rather than x itself, such as y being proportional to x² or x³. To find the constant k, substitute one known pair of values into y=kxⁿ and solve for k, then use the completed formula to find other values. If y is directly proportional to x³ and y=16 when x=2, then 16=k(2³)=8k, so k=2, giving y=2x³. Once k is known, doubling x in y=kx² quadruples y, since (2x)²=4x², showing how the power affects scaling. Always identify the power n correctly from the wording, such as "proportional to the square" meaning n=2.

Worked example

y is directly proportional to the square of x. When x=3, y=45. Find y when x=5.

Diagram of a square
  1. 1Write y = kx², substitute x=3, y=45: 45 = k(3²) = 9k
  2. 2Solve for k: k = 45÷9 = 5
  3. 3Formula is y = 5x²
  4. 4Substitute x=5: y = 5(5²) = 5×25 = 125

Answer: 125

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. Pupils assume doubling x always doubles y, ignoring the power. For y=kx², doubling x actually multiplies y by 4, not 2, since (2x)²=4x². Always apply the power to the scale factor, not just multiply by it directly.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard