The rate of change is the change in output divided by the change in input.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
A line is special
For a straight line the rate is the same everywhere. That number is the slope.
A curve is not
For a curve the rate depends on the interval, so the interval must always be named.
Say the units
"3" means nothing; "3 pounds per hour" is an answer someone can act on.
Average rate of change
Between two points, the average rate of change is the change in output over the change in input — the slope of the line joining them. For a straight line this is the same everywhere; for a curve it depends on which two points you choose.
Its units say what it measures
Output units divided by input units: miles per hour, dollars per item, degrees per minute. Stating the units turns a number into a statement about the situation, which is usually what a question is really after.
Constant rate means linear
A function whose average rate of change is the same over every interval is linear. Checking a few intervals in a table is the standard test, and a single mismatched interval rules linearity out.
What the sign tells you
Positive means increasing over that interval, negative means decreasing, zero means the endpoints are level. Over a curve, a zero average rate does not mean nothing happened in between — it may have risen and fallen back.
Step 2: Try It Yourself
Tap and try it out.
Both triangles give the same slope, 2, even though one is bigger. They are the same shape, so their sides are in the same ratio.
Step 3: Watch an Example
One step at a time.
Watch Dev Read a Rate
A tank holds 50 litres at 0 minutes and 30 litres at 4 minutes.
- Step 1
The change in output is 30 − 50 = −20 litres.
Step 4: Your Turn
Practice makes it stick.
The Drain
Problem 1 of 2
From 40 litres to 10 litres over 6 minutes. What is the rate, in litres per minute?
The Wage
Problem 2 of 2
£60 for 5 hours. What is the rate, in pounds per hour?
Per One Unit
1 of 8
From 2 to 14 over 4 steps. Rate?
2 of 8
From 20 to 8 over 3 steps. Rate?
3 of 8
From 7 to 7 over 5 steps. Rate?
4 of 8
Is the rate of change of a straight line constant? 1 for yes, 0 for no.
5 of 8
Through (0, 3) and (4, 19). Rate?
6 of 8
A negative rate means the output is doing what? 1 for rising, 2 for falling.
7 of 8
Select every statement that is true of the rate of change.
8 of 8
Through (1, 5) and (3, 11). Rate?
Step 5: Quick Check
Show what you know.
Question 1 of 1
From 5 to 25 over 4 steps. Rate?
What You Learned
- Rate of change is the change in output divided by the change in input.
- For a line it is constant and equals the slope; for a curve it depends on the interval.
- A rate without units is not an answer.