A linear function changes by the same amount for every equal step in x.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
On a graph
A linear function is a straight line. A curve of any kind is not linear.
In a table
Check that equal steps in x give equal steps in y. 2, 5, 8, 11 rises by 3 each time, so it is linear.
In an equation
y = mx + b is linear. An x², a 1/x, or an x in an exponent is not.
Straight or not
A linear function graphs as a straight line; anything else is nonlinear. The distinction is visible immediately on a graph, which is one reason graphs are drawn before equations are analysed.
Spotting it in a table
A linear function has a constant rate of change: equal steps in x produce equal steps in y. If the differences vary, it is nonlinear. Checking differences down a table is a quick, reliable test.
Spotting it in an equation
Linear functions have x to the first power only — no x², no 1/x, no √x. y = 3x + 1 is linear; y = x² is not. The form of the equation gives the answer at a glance.
Which situations are which
A constant hourly wage is linear. Compound interest, area against side length, and falling objects are not. Assuming linearity when a situation is not is a serious modelling error, and it is one people make constantly.
Step 2: Try It Yourself
A straight line, for comparison with the curves you will meet later.
The slope is 2: for every 1 across, the line goes 2 up.
Step 3: Watch an Example
One step at a time.
Watch Ana Test Two Tables
Table A: 3, 7, 11, 15. Table B: 1, 4, 9, 16.
- Step 1
Table A rises by 4 each time, so the rate is constant.
Step 4: Your Turn
Practice makes it stick.
The Table
Problem 1 of 2
5, 9, 13, 17. What is the constant rise?
The Curve
Problem 2 of 2
Is y = x² linear? 1 for yes, 0 for no.
Straight or Not
1 of 4
2, 6, 10, 14. Constant rise?
2 of 4
Is y = 3x + 2 linear? 1 for yes, 0 for no.
3 of 4
1, 2, 4, 8. Is it linear? 1 for yes, 0 for no.
4 of 4
10, 7, 4, 1. Constant rise?
Step 5: Quick Check
Show what you know.
Question 1 of 1
4, 10, 16, 22. Constant rise?
What You Learned
- A linear function changes by the same amount for every equal step in x.
- Its graph is a straight line and its equation has the form y = mx + b.
- A changing rate of change means the function is not linear.