How to Identify Linear vs. Nonlinear Functions? (+FREE Worksheet!)

Not every relationship in mathematics follows a straight line. Some relationships curve, bend, or grow at an ever-increasing rate. Knowing how to tell a linear function from a nonlinear function is one of the most important skills in 8th-grade math. Once you can classify a function, you unlock the right set of tools—slope-intercept form for lines, or more advanced techniques for curves.

In this comprehensive guide, you will learn three fast tests (table test, equation test, and graph test), walk through detailed examples, and practice with problems and full solutions.

What Makes a Function Linear?

A function is linear if it produces a straight-line graph. Equivalently:

• Its equation can be written in the form \(y = mx + b\), where \(m\) and \(b\) are constants and the variable \(x\) has an exponent of exactly 1.
• In a table of values, equal changes in \(x\) always produce equal changes in \(y\) (constant rate of change).
• Its graph is a straight line.

What Makes a Function Nonlinear?

A function is nonlinear if it does not satisfy the conditions above. Common nonlinear functions include:

• Quadratic: \(y = ax^{2} + bx + c\) (parabola)
• Exponential: \(y = a \cdot b^{x}\) (rapid growth or decay)
• Absolute value: \(y = |x|\) (V-shape)
• Square root: \(y = \sqrt{x}\) (curve that flattens out)

Three Tests to Classify a Function

Test 1 — The Table Test (First Differences)

1. Make sure the \(x\)-values increase by a constant amount (e.g., 1, 2, 3, 4 …).
2. Calculate the first differences: \(\Delta y = y_{n+1} – y_{n}\) for each consecutive pair.
3. If every first difference is the same, the function is linear. If they vary, it is nonlinear.

Test 2 — The Equation Test

Look at the equation:

• If \(x\) (or the input variable) appears only to the first power and is not inside a radical, absolute value, or exponent, the function is linear.
• Any higher power (\(x^{2}\), \(x^{3}\)), product of variables (\(xy\)), root (\(\sqrt{x}\)), or base with a variable exponent (\(2^{x}\)) makes it nonlinear.

Test 3 — The Graph Test

Plot the points or look at the given graph:

• Straight line → linear.
• Any curve (parabola, exponential curve, V-shape, S-curve) → nonlinear.

Worked Examples

Example 1 — Table Test

All first differences are \(4\). Linear. The equation is \(y = 4x – 1\).

Example 2 — Table Test (Nonlinear)

First differences are \(3, 5, 7\)—not constant. Nonlinear (this is \(y = x^{2}\)).

Example 3 — Equation Test

Classify each equation:

• \(y = 5x – 9\) → \(x\) to the first power only. Linear.
• \(y = x^{2} + 3\) → \(x^{2}\) term. Nonlinear.
• \(y = \frac{2}{x}\) → \(x\) in the denominator = \(2x^{-1}\). Nonlinear.
• \(y = 3 \cdot 2^{x}\) → variable in the exponent. Nonlinear.

Example 4 — Real-World Context

A ball is dropped from a rooftop. Its height (in feet) after \(t\) seconds is \(h = -16t^{2} + 64\). Is this linear or nonlinear?

The \(t^{2}\) term makes this nonlinear—the ball accelerates due to gravity, falling faster each second.

Video Lesson

Watch this video for additional examples and a step-by-step walkthrough:

Comparing Linear, Quadratic, and Exponential Growth

Practice Problems

1. Is \(y = 7x + 2\) linear or nonlinear?
2. Is \(y = x^{3} – 1\) linear or nonlinear?
3. A table has \(x = 1, 2, 3, 4\) and \(y = 2, 4, 8, 16\). Linear or nonlinear?
4. A table has \(x = 0, 1, 2, 3\) and \(y = 5, 8, 11, 14\). Linear or nonlinear?
5. Is \(y = |x – 3|\) linear or nonlinear?
6. Classify: \(y = -\frac{1}{2}x + 10\).
7. Classify: \(y = 4^{x}\).
8. A car travels at a constant 60 mph. Is the distance-time function linear or nonlinear?
9. A population doubles every year starting at 100. Is this linear or nonlinear?
10. A table has \(x = 1, 2, 3, 4\) and \(y = 1, 4, 9, 16\). Linear or nonlinear? What type of function is this?

Solutions

1. Linear. The equation is in \(y = mx + b\) form.
2. Nonlinear. The \(x^{3}\) term has degree 3.
3. Nonlinear. Differences: \(2, 4, 8\)—not constant. (Exponential: \(y = 2^{x}\).)
4. Linear. Differences: \(3, 3, 3\). Equation: \(y = 3x + 5\).
5. Nonlinear. The absolute-value creates a V-shaped graph.
6. Linear. Slope \(-\frac{1}{2}\), y-intercept 10.
7. Nonlinear. Variable in the exponent (exponential).
8. Linear. \(d = 60t\), constant rate of change.
9. Nonlinear. \(P = 100 \cdot 2^{t}\), exponential growth.
10. Nonlinear. Differences: \(3, 5, 7\). This is quadratic: \(y = x^{2}\).

Common Mistakes to Avoid

• Assuming any equation with an \(x\) is linear. Check the exponent: \(x^{2}\), \(\sqrt{x}\), and \(2^{x}\) are all nonlinear.
• Forgetting to verify constant \(\Delta x\) in a table. If the \(x\)-values are not equally spaced, first differences alone will not tell you if the function is linear. Compute the slope between each pair of points instead.
• Confusing “not a function” with “nonlinear.” A vertical line is not a function at all. A curve like a parabola is a function—just a nonlinear one.

Frequently Asked Questions

Can a nonlinear function have a constant section?

Yes. A piecewise function might be linear over one interval and curved over another. Classify the entire function by its overall behavior.

What is the fastest way to tell on a test?

If you see the equation, look for \(x^{2}\), \(x^{3}\), \(\sqrt{x}\), \(|x|\), or \(b^{x}\)—any of these make it nonlinear. If you only have a table, compute the first differences and check for a constant value.