Functions
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Introduction to Non-Linear Functions

Linear vs quadratic vs exponential — three curves, three stories

Not every function is a straight line. A linear function like y = 2x grows at a constant rate. A quadratic like y = x^2 grows faster and faster. An exponential like y = 2^x grows faster than either — eventually it dwarfs everything else.

Understanding the difference matters in real life: a salary that grows linearly adds the same raise each year; an investment that grows exponentially doubles on a schedule; a ball thrown upward follows a quadratic path.

In this lesson, you'll see all three on one graph and discover what makes each one unique.

Graph

FAQ

What is a non-linear function?
A non-linear function is any function whose graph is not a straight line. Quadratics (parabolas), exponentials, square roots, and trigonometric functions are all non-linear. The rate of change is not constant — it speeds up, slows down, or oscillates.
How does quadratic growth differ from linear?
A linear function adds the same amount each step (constant rate of change). A quadratic adds more and more each step — the rate of change itself increases linearly. For example, at x = 1, x² = 1; at x = 10, x² = 100; the jumps get bigger.
Why does exponential growth beat everything?
An exponential function multiplies by a constant factor each step, so it doubles, then doubles again. Eventually it overtakes any polynomial. At x = 10: 2x = 20, x² = 100, 2^x = 1024. At x = 20: 2x = 40, x² = 400, 2^x = 1,048,576.
When do I use each type of function?
Use linear for constant-rate situations (flat salary increase, constant speed). Use quadratic for acceleration (free fall, area). Use exponential for compound growth (interest, population, radioactive decay).

Related Topics

Linear EquationsQuadratic EquationsExponential Growth & Decay