Nonlinear Relationship Math Example 1

Follow the full solution, then compare it with the other examples linked below.

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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?

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

About Nonlinear Relationship

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

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

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

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