Linear Functions vs Nonlinear Functions

Product description

From the author

This worksheet helps students to connect graphs and equations and distinguish linear functions from nonlinear functions. The aim is to understand that the equation y = mx + b defines a linear function, whose graph is a straight line. This is for common core course 8.F.A.3.

How to use: The three graph-matching pages after the introduction have 6 graphs and a box at the bottom of the page. Students should pick equations from the box and connect them to the right graph. After that, they have to decide whether the graph-equation pair is a linear function.

After that, there is a page of concept understanding, where students have to give verbal answers to the following questions: What do linear functions look like in a coordinate plane? Can you think of any situation when a straight line in a coordinate plane is not a function? Give three examples of equations of linear functions. Give two examples of equations of nonlinear functions.

Along obviously linear and nonlinear functions, there is a task where the function is piecewise linear. (The students are supposed to answer that it's not linear, and in the answers section the concept of piecewise linear is explained.)

There are 4 task pages and 2 answer pages in this resource.

In the classroom

Teaching tips & learning objectives

Learning objectives

Distinguish linear functions from nonlinear functions given an equation
Explain why a vertical line is not a function
Write examples of linear functions in y = mx + b form and separate examples of nonlinear functions

Teaching tips

Assign after a lesson introducing the y = mx + b form so students can apply, not just learn, the linear-function definition.
Discuss the y = |x| answer explanation as a class, since the file itself flags it as a common misconception (piecewise but not linear).
Use the four explain-your-reasoning questions as a bell-ringer or exit ticket before assigning the matching sections.

Skills covered

Linear vs. nonlinear classification — students match equations like y = x - 6 and y = 3x squared + 1 to graphs and decide whether each is linear.
Function reasoning — students explain when a straight line, such as x = 2, fails to be a function.
Constructing examples — students write original linear-function equations in y = mx + b form and provide separate nonlinear examples.

Good to know

Questions teachers ask about this resource

Why is y = |x| classified as nonlinear?

Its graph changes direction at a corner. A linear function has one straight-line graph throughout.

Why is x = 2 not a function of x?

It is a vertical line with more than one y-value for the same x-value, so it does not assign a single output to each input.

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Curriculum

Standards, concepts & topics

Standards

8.F.A.3 — Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

Core concepts

Linear functions
Nonlinear functions
Coordinate graphing
Function equations

Topics

Y=mx+b Linear Nonlinear Connect graph and equation Algebra

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