Linear vs. Nonlinear Functions
Aligned to 8.F.A.3 — Common Core State Standards for Mathematics.
What this lesson teaches
A function in the form y = mx + b is linear — its graph is a straight line with a constant rate of change m. Functions that don't fit this form, like y = x^2, are nonlinear and graph as curves.
Worked examples
Example 1
y = 3x − 1 is linear (m = 3, b = −1); equal steps in x give equal steps in y. But y = x^2 is nonlinear: at x = 1, 2, 3 the outputs are 1, 4, 9, and the differences 3, 5, 7 are not constant.
Example 2
y = 2/x is nonlinear: at x = 1, 2, 4 the outputs are 2, 1, 0.5, dropping by different amounts each time.
Example 3
A straight line on a graph and a constant difference in a table are the same fact seen two ways.
Practice questions
- Is y = −2x + 7 linear or nonlinear? Give its slope and intercept.
- Explain why y = x^2 + 1 is not linear.
- Make a table for y = 2^x at x = 0, 1, 2, 3 and explain why it is nonlinear.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Funciones Lineales y No Lineales
Una función de la forma y = mx + b es lineal — su gráfica es una recta con una tasa de cambio constante m. Las funciones que no tienen esta forma, como y = x^2, son no lineales y se grafican como curvas.
Ejemplo 1: y = 3x − 1 es lineal (m = 3, b = −1); pasos iguales en x dan pasos iguales en y. Pero y = x^2 es no lineal: en x = 1, 2, 3 las salidas son 1, 4, 9, y las diferencias 3, 5, 7 no son constantes.
Ejemplo 2: y = 2/x es no lineal: en x = 1, 2, 4 las salidas son 2, 1, 0.5, y bajan cantidades distintas cada vez.
Ejemplo 3: Una recta en la gráfica y una diferencia constante en la tabla son el mismo hecho visto de dos formas.
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