Nonlinear Relationship Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Nonlinear Relationship.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A relationship between two quantities where the rate of change is not constant—the graph is curved, not a straight line.

Not a straight line—it curves. Compound interest grows faster and faster.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Nonlinear relationships have a rate of change that itself changes—equal inputs don't give equal jumps in output.

Common stuck point: Recognizing that 'constant ratio' (exponential) is still nonlinear.

Sense of Study hint: Compare successive differences in the y-values -- if they change, the relationship is not linear.

Worked Examples

Example 1

medium
The area of a square is \(A = x^2\). Compare how \(A\) changes when \(x\) goes from 1 to 2, then 2 to 3. Is this relationship linear?

Solution

  1. 1
    \(A(1)=1, A(2)=4, A(3)=9\).
  2. 2
    Change from \(x=1\) to \(x=2\): \(\Delta A = 3\).
  3. 3
    Change from \(x=2\) to \(x=3\): \(\Delta A = 5\).
  4. 4
    The changes are not equal (3 ≠ 5), so the rate of change is not constant.
  5. 5
    This is a nonlinear (quadratic) relationship.

Answer

Not linear — the rate of change increases
A linear relationship has a constant rate of change. Here \(\Delta A\) grows with \(x\), so \(A = x^2\) is nonlinear (quadratic).

Example 2

hard
For \(f(x) = x^2 - 4x + 3\), find the vertex and determine whether the parabola opens up or down.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
For \(y = x^2\), calculate values at \(x = -2, -1, 0, 1, 2\). Is the graph symmetric?

Example 2

hard
Determine whether \(y = 2^x\) is linear or nonlinear by examining the ratio \(y_{n+1}/y_n\) for \(x = 0, 1, 2, 3\).
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Background Knowledge

These ideas may be useful before you work through the harder examples.

linear relationship