Part 11 of 20 · LinesSee the whole course
How to tell if a function is linear
A function is linear when its output changes by the same amount for every 1 step of the input. In a table, work out the change in y between neighboring rows. If every change is the same, the function is linear, and that steady change is the slope.
Steady change makes a straight line
A line is what you get when something changes by the same amount every step. A taxi that charges 3 to start and 2 per kilometer costs 3, 5, 7, 9 for 0, 1, 2, 3 kilometers. Every kilometer adds 2, always 2. That is linear.
A rising graph is not enough. A graph can climb faster and faster and still be a curve. Steady change is what counts.
The table test
- Look at the x-values. Do they go up by the same step each row?
- Find the change in y between each pair of rows: the next y minus this y.
- Divide by the change in x. If the x-values go up by 1 each time, there is nothing to divide by.
- Compare. If every answer is the same number, the function is linear, and that number is its slope.
Try it below. Pick a table and look at the “rate” row.
Example 1: a table that passes
x: 0, 1, 2, 3
y: 1, 3, 5, 7
Changes in y: 3 − 1 = 2, 5 − 3 = 2, 7 − 5 = 2
Always 2, so it is linear, with slope 2: y = 2x + 1
Example 2: tables that fail
y: 1, 2, 4, 8
Changes: 1, 2, 4. Not the same, so not linear (it keeps doubling).
y: 0, 1, 4, 9
Changes: 1, 3, 5. Not the same, so not linear (these are squares).
When the x-values skip
If the x-values do not go up by 1, you must divide each change in y by the change in x. Otherwise a bigger jump in x fools you into thinking the function is speeding up.
x: 0, 1, 3
y: 1, 3, 7
0 to 1: (3 − 1) ÷ (1 − 0) = 2
1 to 3: (7 − 3) ÷ (3 − 1) = 4 ÷ 2 = 2
Both 2, so it is linear.
The change in y between 1 and 3 is 4, which looks bigger than 2, but x moved 2 steps, so it is still 2 per step.
From a graph or a story
- A graph is linear when it is a straight line.
- A story is linear when the same amount is added for each unit. Saving 50 each week is linear. A population that doubles each year is not.
Watch it explained in under a minute
The five mistakes to avoid
- Calling any rising table linear. Rising is not enough: the steps must be equal.
- Forgetting uneven x-values. Divide each change in y by the change in x.
- Checking only the first two rows. Check every step.
- Comparing the y-values instead of the changes between them.
- Mixing up a total with a rate. “50 saved so far” is a total, “50 more each week” is the rate.
Try it yourself
Is each table linear? Tap a question to see the answer.
1. x: 1, 2, 3, 4 and y: 5, 8, 11, 14
The changes are 3, 3, 3, so it is linear, with slope 3.
2. x: 1, 2, 3, 4 and y: 2, 4, 8, 16
The changes are 2, 4, 8, so it is not linear.
3. x: 0, 2, 6 and y: 1, 5, 13
(5 − 1) ÷ 2 = 2 and (13 − 5) ÷ 4 = 2. Both are 2, so it is linear.
Where this fits in
The steady change is the slope, so this ties together what slope is and slope-intercept form. Next in this course: turning a story into y = mx + b.