Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for 8th grade functions? Then this Grade 8 lesson on Understanding Functions and Their Graphs explains each idea step by step, with worked examples.
Quick answer: A function assigns exactly one output to every input. You can check it in a table, a mapping, a graph (vertical line test) or an equation. Linear functions, y = mx + b, change at a constant rate and graph as straight lines; nonlinear functions such as A = s² do not.
Standards: 8.F.A.1, 8.F.A.2, 8.F.A.3 · Level: Grade 8
What is a function?
A function is a rule that assigns exactly one output to each input. If an input could produce two different outputs, the relation is not a function.
We write f(x) for the output of the function f at input x. For f(x) = 3x + 2, we have f(4) = 14.
How do you test whether a relation is a function?
| Representation | Check |
|---|---|
| Ordered pairs or table | No input appears with two different outputs |
| Mapping diagram | Every input has exactly one arrow |
| Graph | Vertical line test: no vertical line crosses the graph twice |
| Equation | Each value of x gives only one value of y |
Two inputs may share an output. A function can send 2 and −2 to the same value, but it cannot send one input to two values.
How do you tell linear functions from nonlinear ones?
A linear function has the form y = mx + b. Its rate of change m is constant, so its graph is a straight line. If the rate of change varies, the function is nonlinear.
For the area of a square, the values 1, 4, 9 grow by 3, then by 5. The rate is not constant, so A = s² is nonlinear.
- Get the rate of change and initial value of each from its table, graph or equation.
- Compare rates: the larger rate grows faster.
- Compare initial values: the value when x = 0.
Function A: y = 5x + 2. Function B: a table with (0, 9) and (2, 17). B has rate (17 − 9) ÷ 2 = 4 and initial value 9.
A grows faster (5 > 4), but B starts higher (9 > 2).
Thinking repeated outputs break the rule. The rule only forbids one input having two outputs, so f(2) = f(−2) = 4 is fine.
Beep! In a table, check the differences. If equal steps in x give equal steps in y, the function is linear.
- Function: every input has exactly one output.
- Vertical line test: a graph is a function if no vertical line hits it twice.
- Different inputs may share the same output.
- Linear function: y = mx + b with constant rate of change m.
- Compare functions by rate of change and initial value, whatever their representation.
Understanding Functions and Their Graphs: frequently asked questions
What is the difference between a relation and a function?
A relation is any set of input-output pairs. A function is a relation where each input has exactly one output, so {(1, 2), (1, 5)} is not a function.
How does the vertical line test work?
Slide a vertical line across the graph. If it never touches the graph in more than one point, each input has one output and the graph is a function.
Can two different inputs have the same output?
Yes. For f(x) = x², both 3 and −3 give 9. That is allowed, because the rule only restricts one input from having two outputs.
How can you tell if a function is linear from a table?
Compute the change in y for equal steps in x. If the rate of change is constant, the function is linear; if it varies, it is nonlinear.
What does f(5) mean?
It means the output of the function f when the input is 5. For f(x) = 4x − 7, substitute 5 for x to get f(5) = 13.