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 = s 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.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Any equation that can be written as y = mx + b describes a linear function: a constant rate of change m, a starting value b, and a graph that is a straight line. Eighth graders connect these three facts and use them to classify functions. In a table with equally spaced inputs, a linear function shows the outputs changing by the same amount every step.
Just as important is recognizing what is not linear. The area of a square as a function of its side length, A = s², produces the pairs (1, 1), (2, 4), (3, 9). The jumps of 3 then 5 are unequal, and the points curve rather than line up. Other common nonlinear examples include y = 2ˣ (doubling), y = 12/x and y = x³. Students learn to spot the warning signs in an equation (a variable squared, in an exponent or in a denominator) and in tables and graphs, and to give their own examples of each kind.
If the inputs in a table go 1, 2, 4, 8, equal output jumps do not mean a constant rate. Divide each change in y by the change in x.
A horizontal line has rate of change 0 and is still linear: y = 0x + 5. It also passes the vertical line test.
Equations like 3x + y = 7 do not look like y = mx + b, but rearranging gives y = -3x + 7, so they are linear.
A graph can rise steadily and still curve, like y = x² for positive x. Linear means a constant rate, which shows as a straight line.
Is the function shown linear? x: 0, 1, 2, 3, 4 and y: 1, 3, 7, 13, 21.
Answer: Nonlinear, because the output changes by 2, 4, 6 and 8 rather than by a constant amount.
Have students graph y = 2x, y = x² and y = 2ˣ on the same axes from tables they build themselves. The comparison makes the shapes memorable and gives a reason why the constant difference test works. Pattern growth tasks (squares made from tiles, stairs made from blocks) offer natural nonlinear examples.
Assessment items often ask which equation, table or graph represents a nonlinear function. Teach students to name the evidence they used: a squared or exponent variable, unequal differences, or a curved graph. This also prepares them for distinguishing linear and exponential models in high school.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: C) y = x² + 3
y = x² + 3 contains x squared, so its graph is a curve. The others can all be written as y = mx + b.
Answer: B) x: 1, 2, 3 and y: 5, 8, 11
In y: 5, 8, 11 the output rises by 3 each time the input rises by 1, a constant rate of change.
Answer: 3
Add 6x: 2y = 6x + 4. Divide by 2: y = 3x + 2. The slope is 3.
Answer: D) Because the points (1, 1), (2, 4), (3, 9) do not lie on a straight line
The changes 3 and 5 between consecutive points are unequal, so the graph curves rather than forming a straight line.
Answer: 19
The rate is (10 - 4) ÷ 2 = 3, so y = 3x + 4. When x = 5, y = 15 + 4 = 19.
Answer: B) Bacteria doubling every hour
Doubling multiplies by 2 each hour, so the amount grows by larger and larger steps. The other situations change by a constant amount.
Distinguish between situations that can be modeled with linear functions and with exponential functions.
A full lesson with slides, activities and an exit ticket on linear and nonlinear functions, pitched to grade 8 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 8.F.A.3 with an answer key, ready in about a minute.
Make a worksheet →Turn linear and nonlinear functions into a quiz students answer online that marks itself, with a class summary for you.
Build a test →Every non-vertical straight line is. A vertical line such as x = 3 is not a function at all, because one input has infinitely many outputs.
They need to recognize that a function is not linear and give examples. Names such as quadratic and exponential are helpful but are studied fully in high school.