Linear Functions • Topic 2 of 7

Rate of Change

For a linear relationship, the rate of change is the slope: how much the output changes per unit increase in the input, and it stays constant throughout. In a real context it is the "per" quantity — dollars per hour, miles per gallon, growth per year. From a table, divide the change in the output by the change in the input between any two rows; if that ratio is constant, the relationship is linear. The SAT often phrases slope as a rate ("the cost increases by $3 each month"), so recognising rate-of-change language as slope is key.

✅ Solved examples

1. A plan charges $5 per month. What is the rate of change of cost with time?
The cost rises $5 each month, so the rate of change (slope) is 5.
2. A table shows y = 2, 5, 8 as x = 0, 1, 2. Rate of change?
y increases by 3 each time x increases by 1, so the rate is 3.
3. Water drains 4 litres per minute. Rate of change of volume?
Volume decreases 4 L each minute, so the rate is −4.
4. A car covers 60 miles each hour. Rate of change of distance?
60 miles per hour, so the rate (slope) is 60.

✏️ Practice — try these, take hints as needed

1. A gym charges $8 per visit. Rate of change of cost with visits?
Cost rises $8 per visit.
That per-unit rate is the slope.
—
8.
2. A table: y = 1, 4, 7 at x = 0, 1, 2. Rate of change?
Change in y per change in x.
y rises by 3 each step.
—
3.
3. A tank loses 6 gallons per hour. Rate of change of volume?
Decreasing → negative.
6 per hour.
—
−6.
4. A printer adds 15 pages per minute. Rate of change of pages?
Per-minute increase.
That is the slope.
—
15.
5. A table: y = 10, 16, 22 at x = 0, 2, 4. Rate of change?
Change in y ÷ change in x.
6 over 2.
Compute.
3.

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
← Slope Slope Formula →