Nonlinear Function

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Nonlinear Function: meaning, method and solved example (8th Grade) – Maths PDF

In short: Nonlinear function describes a rule whose output moves by uneven amounts when the input moves in equal steps, which bends its graph away from a straight line.

8th GradeFunctions and Graphs8.F.A.3

What is Nonlinear Function?

Any nonlinear function has a rate of change that refuses to stay put. Consider square tiles with side \(s\) and area \(A = s^2\). Sides 1, 2, 3, 4 yield areas 1, 4, 9, 16, and the jumps between neighbors are 3, 5, 7, still growing.

With unequal jumps, no single slope describes the whole graph and the points fail to line up. Typical sources include squares, cubes, repeated doubling and rules with the variable in a denominator or an exponent.

How to spot it

  • A table shows output gaps that change as the input rises by equal steps.
  • The formula has x squared, x cubed, a variable exponent or a variable in a denominator.
  • A graph bends, curves upward, or levels off instead of running straight.
  • A story involves area, doubling, or a quantity that speeds up or slows down.

The method

  1. Make a table with evenly spaced inputs and subtract each output from the next one.
  2. All differences equal means linear; differences that vary mean nonlinear.
  3. Cross-check with the formula: an x² term or a variable exponent signals a curve.
  4. Describe the behavior from the changing gaps, such as climbing faster and faster.
(1, 1)(2, 4)(3, 9)(4, 16)A = s²dashed: a line would look like thisside s
The curve A = s² bends away from the dashed straight line.

Nonlinear Function: solved example

Problem

A mosaic artist logs the face area of square tiles: a 1-inch tile covers 1, a 2-inch tile covers 4, a 3-inch tile covers 9 and a 4-inch tile covers 16 square inches. Does a nonlinear function describe this pattern?

Solution

The side grows by 1 inch each step, while the area grows by \(4 – 1 = 3\), then \(9 – 4 = 5\), then \(16 – 9 = 7\).

Unequal gains mean the rate is not constant. So \(A = s^2\) is a nonlinear function and its picture is a curve.

Try it

Question 1

A bacteria count doubles each hour: 5, 10, 20, 40 at hours 0 to 3. Linear or nonlinear?

Show the answer

The gaps are 5, 10, 20, which keep growing, so it is nonlinear.

Question 2

Which is nonlinear: \(y = 4x – 1\) or \(y = \frac{12}{x}\)?

Show the answer

\(y = \frac{12}{x}\) has a variable in the denominator, so its graph is a curve; \(y = 4x – 1\) is a line.

Nonlinear Function vs. Linear Function

A nonlinear function breaks the one rule that defines a linear one: equal input steps no longer produce equal output changes. It is still a function, with exactly one output for each input. Nonlinear describes the shape of the graph, not the validity of the rule.

Linear Function

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