Linear Function
In short: Linear function names a rule f(x) = mx + b that draws a straight line, since every equal step in the input moves the output by one fixed amount.
8th GradeFunctions and Graphs8.F.A.3
What is Linear Function?
Every linear function runs on a constant rate. Raise the input by 1 and the output goes up, or down, by the same \(m\) each time, while \(b\) records the output at \(x = 0\).
That steadiness explains the formula \(f(x) = mx + b\) and the straight graph. A cab charging \(\$4\) at pickup plus \(\$2.50\) per mile follows \(f(x) = 2.5x + 4\). In a table, such rules show identical output gaps for identical input gaps.
How to spot it
- A table repeats the same output difference whenever the input climbs by the same step.
- A story offers a flat starting amount plus a fixed charge per unit, like a fare or a fee.
- The graph runs straight, or the rule can be rearranged into y = mx + b.
- The variable is never squared, never in a denominator and never in an exponent.
The method
- Test steadiness: equal input steps should bring equal output changes, or the formula should fit mx + b.
- Get the rate \(m\) by dividing the output change by the input change.
- Get the starting value \(b\), the output at input 0, and assemble \(f(x) = mx + b\).
- To evaluate, plug in an input; to reverse, solve \(mx + b = k\) for the input matching a given output.
Linear Function: solved example
Problem
A taxi charges a \(\$4\) pickup fee plus \(\$2.50\) per mile, so the fare is \(f(x) = 2.5x + 4\). What does a 10-mile ride cost, and how far can a rider go for \(\$54\)?
Solution
Ten miles: \(f(10) = 2.5(10) + 4 = 29\) dollars.
For \(\$54\): \(2.5x + 4 = 54\), so \(2.5x = 50\) and \(x = 20\). The ride costs \(\$29\), and \(\$54\) covers 20 miles.
Try it
Question 1
Inputs 0, 1, 2, 3 produce outputs 5, 9, 13, 17. Does a linear function fit the table? State its rule.
Show the answer
The outputs rise by 4 each time, so yes. Starting at 5 with rate 4: \(f(x) = 4x + 5\).
Question 2
Is \(y = x^2\) a linear function?
Show the answer
No. The outputs 0, 1, 4, 9 are separated by gaps of 1, 3, 5, which are unequal, and the picture bends.
Linear Function vs. Nonlinear Function
A linear function changes by a constant amount per step, so its graph is straight. A nonlinear function changes by varying amounts, so its graph bends, as with areas of squares or doubling populations. Checking the gaps in a table is the quickest way to tell.
Keep going
Lessons
Worksheets and quizzes
- Linear Functions as Models – Worksheet
- Linear Functions as Models – Quiz
- Writing Linear Equations and Functions – Worksheet
- Writing Linear Equations and Functions – Quiz
Related words
Whole dictionary: A–Z · 8th Grade · Functions and Graphs