Take the average rate of change over successive equal intervals. Whether those rates rise or fall is concavity.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Concave up
The rates are increasing. The curve bends upward, like the inside of a bowl.
Concave down
The rates are decreasing. The curve bends downward, like a dome.
Concavity is not direction
A function can be decreasing and concave up at the same time — falling, but flattening out.
The rate of the rate
Concavity describes whether the rate of change is itself increasing or decreasing. Concave up means the rate is increasing; concave down means it is decreasing. A function can be increasing while concave down.
Four combinations, all possible
Increasing and concave up, increasing and concave down, decreasing and concave up, decreasing and concave down. Direction and curvature are independent, and describing both is what the exam asks for.
Reading concavity from a table
Compute average rates over successive equal intervals. If those rates are increasing, the function is concave up. This turns a curvature question into an arithmetic one, which is how it usually appears in data.
The language matters
"Increasing at a decreasing rate" describes an increasing, concave-down function. That phrasing is standard on the exam and in economics. Being able to translate between the phrase and the graph is a specific, testable skill.
Step 2: Try It Yourself
Tap and try it out.
- Point(-3, -27)
- Second point(-1, -1)
- Slope between them13
Slide the second point towards the first. The slope between them approaches the slope of the curve at that point.
Step 3: Watch an Example
One step at a time.
Watch Sam Test a Parabola
Sam checks the concavity of f(x) = x² using three equal intervals.
- Step 1
On [0, 1] the rate is (1 − 0) ÷ 1 = 1.
Step 4: Your Turn
Practice makes it stick.
Successive Rates
Problem 1 of 2
Successive rates over equal intervals are 8, 5, 2. What is the concavity? Answer 1 for up, 2 for down.
Falling but Flattening
Problem 2 of 2
Rates over equal intervals are −9, −4, −1. Is the function increasing? Answer 1 for yes, 0 for no.
Read the Bend
1 of 8
Rates 2, 4, 6. Concavity? 1 for up, 2 for down.
2 of 8
Rates 6, 3, 0. Concavity? 1 for up, 2 for down.
3 of 8
Rates −6, −3, 0. Concavity? 1 for up, 2 for down.
4 of 8
Sort each description into the concavity it describes.
Tap something to move it.
- Empty
- Empty
5 of 8
f(x) = x² on [0,1], [1,2], [2,3]. What is the third rate?
6 of 8
Can a function be decreasing and concave up at once?
7 of 8
Rates 4, 4, 4. Concavity? 0 for neither, 1 for up, 2 for down.
8 of 8
A cubic changes concavity how many times?
Step 5: Quick Check
Show what you know.
Question 1 of 1
Rates over equal intervals are 1, 5, 11. Concavity? 1 for up, 2 for down.
What You Learned
- Concavity is what the average rates of change are doing over successive intervals.
- Rising rates mean concave up; falling rates mean concave down.
- Concavity and direction are independent — a function can fall while flattening.