8.F.3: Linear and Nonlinear Functions
Also written as 8.F.A.3 and CCSS.MATH.CONTENT.8.F.A.3 · Cluster: Define, evaluate, and compare functions.
The Official Wording
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.
Common Core State Standards, Grade 8, Functions · Read it at the source · Checked October 14, 2026
What 8.F.3 Means in Plain English
Eighth graders learn that y = mx + b always graphs as a straight line: y = 3x − 2 climbs 3 for each step of 1 in x. Then they meet functions that bend, like a cube’s volume, V = s³, which goes 1, 8, 27.
The test students carry away is equal steps: the same change in x always brings the same change in y. That check feeds 8.F.4, where they build linear models, and high school, where linear growth is set against exponential growth (HSF-LE.1).
8.F.3 I Can Statements
- I can explain why y = mx + b graphs as a straight line.
- I can test a table for equal steps to decide if it is linear.
- I can give an example of a function that is not linear.
8.F.3 Worked Example
Two tables that start and end on the same points, so only the steps in between tell them apart.
Table A: x = 0, 1, 2, 3 gives y = 1, 4, 7, 10. Table B: the same x values give y = 1, 2, 5, 10. Which function is linear? Write its equation.
- The x values go up by 1 each time, so look at how much y changes at each step of Table A.A: 4 − 1 = 3, 7 − 4 = 3, 10 − 7 = 3
- Every step is the same, so A is linear. The constant change is the slope, and y = 1 at x = 0 is where the line starts.A: y = 3x + 1
- Now Table B. Its steps grow, so equal steps in x do not bring equal steps in y.B: 2 − 1 = 1, 5 − 2 = 3, 10 − 5 = 5
- Plot both on one grid. A’s four points sit on one straight line, while B’s points sag below it and curve up to meet it at (3, 10).B fits y = x² + 1, a curve, not a line
Check it: Substitute into A’s equation: 3 × 2 + 1 = 7 and 3 × 3 + 1 = 10 match the table. And the line through (0, 1) and (3, 10) passes through (1, 4), but Table B has (1, 2) instead.
Common Mistakes with 8.F.3
Judging by the first step
“B goes 1, 2, so it adds 1 each time and is linear.”
Try this: Check every step, not just the first. A function is linear only when every equal step in x gives the same change in y.
Thinking a line must pass through the origin
“y = 3x + 1 is not linear because it doesn’t start at 0.”
Try this: Graph y = 3x + 1 beside y = 3x. Both are straight and parallel, and the extra 1 just lifts the line up.
Missing uneven steps in x
x = 1, 2, 4 with y = 5, 7, 11 called nonlinear because y grows by 2, then 4
Try this: Divide each change in y by its change in x. Here 2 ÷ 1 and 4 ÷ 2 both give 2, so it is linear.
Where 8.F.3 Fits
Practice 8.F.3
Quick Check
- Is y = 5 − 2x a linear function? What is its slope?
- A table has x = 0, 1, 2, 3 and y = 2, 4, 8, 16. Linear or not?
- The area of a circle is A = πr². Show it is not linear, using 3.14 for π and r = 1, 2, 3.