Linear Functions vs Nonlinear Functions
- Grades
- Grade 8, Middle School
- File type
- Preparation
- Print ready
- Subject
- Math, Algebra, Graphing, Common Core
- Topic
- y=mx+b, linear
- Resource types
- Worksheets & Printables, Worksheets
- Preparation
- Print ready
- Standards
- 8.F.A.3
About this resource
What's inside this resource
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 first three pages 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.
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In the classroom
Teaching tips & learning objectives
Learning objectives
Distinguish linear functions from nonlinear functions given an equation
Match a function's equation to its correct graph
Explain why a vertical line is not a function
Write example equations for linear and nonlinear functions in the form y = mx + b
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 equations of linear and nonlinear functions in the y = mx + b form.
Good to know
Questions teachers ask about this resource
Why is y = |x| not considered linear even though it looks like two straight lines?
It is piecewise linear, made of two line segments joined at a corner; a linear function's graph must be a single straight line throughout, so y = |x| does not meet that definition despite looking straight in each piece.
How does the worksheet explain why x = 2 is not a linear function?
It notes that x = 2 is a vertical line, which fails the definition of a function because a function must have only one output for each input.
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Curriculum
Standards, concepts & topics
Standards
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
Core concepts
linear functions
nonlinear functions
coordinate graphing
function equations
Topics
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