Nonlinear Relationship Formula

The Formula

y = x^2 (quadratic), y = 2^x (exponential), y = \frac{1}{x} (rational)

When to use: Not a straight line—it curves. Compound interest grows faster and faster.

y = x^2 is nonlinear: 1 \to 1, 2 \to 4, 3 \to 9. The jumps get bigger.

A curved graph indicates a nonlinear relationship; the equation is not of the form y = mx + b

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.

\frac{\Delta y}{\Delta x} \neq \text{const}; \; \text{equivalently, } \frac{f(x_2) - f(x_1)}{x_2 - x_1} \text{ varies with } x_1, x_2

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?

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).

hard

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

Assuming any pattern with a rule must be linear — y = x^2 has a rule but is nonlinear
Confusing 'constant ratio between terms' (exponential/geometric) with 'constant difference' (linear/arithmetic)
Trying to use y = mx + b for data that clearly curves — check if the differences between consecutive y-values are changing

Most real-world relationships are nonlinear—growth, decay, oscillation.

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.

A curved graph indicates a nonlinear relationship; the equation is not of the form y = mx + b

Most real-world relationships are nonlinear—growth, decay, oscillation.

Recognizing that 'constant ratio' (exponential) is still nonlinear.

Before studying the Nonlinear Relationship formula, you should understand: linear relationship.