Take the average rate of change over successive equal intervals. Whether those rates rise or fall is concavity. Rising rates: concave up, like the inside of a bowl. Falling rates: concave down, like a dome.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Concave down
f(x) = √x on [0, 1], [1, 4], [4, 9]: rates 1, 1/3, 1/5. Falling. The curve keeps rising but ever more slowly: a dome, not a bowl.
Concavity is not direction
A function can be decreasing and concave up at the same time: falling, but flattening out. 1/x for x > 0 has rates −1/2, −1/6, −1/12 on [1, 2], [2, 3], [3, 4]: rising rates, so concave up, while falling.
Second differences
For equal steps, the differences of the rates are the second differences. x²: rates 1, 3, 5, second differences 2, 2. Constant and positive: concave up everywhere. A quadratic always has constant second differences.
Inflection
Where the rates stop rising and start falling, or the reverse, concavity changes: an inflection point. x³: rates on [−2, −1], [−1, 0], [0, 1], [1, 2] are 7, 1, 1, 7. Falling then rising: the inflection is at 0.
Not the axis
Never read concavity from whether the graph is above or below the axis. That is a different question. x² is concave up above the axis; −x² + 10 is concave down, also above the axis.
In context
A population growing by more each year: concave up. A car braking, still moving forward but covering less each second: decreasing rates, concave down. Concavity is the rate of the rate.
Step 2: Try It Yourself
Tap and try it out.
Decide the concavity of f(x) = x³ − 6x² on [0, 4] using unit intervals, and locate the inflection
- 1f(0) = 0, f(1) = −5, f(2) = −16, f(3) = −27, f(4) = −32values at each whole number
Step 0 of 4
- 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: In Real Life
A startup’s growth
Users grew by 100, then 150, then 225 a month. The growth is growing: concave up. When the monthly gains start shrinking, the curve turns concave down, and investors notice first.
Step 4: 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. On [1, 2] it is (4 − 1) ÷ 1 = 3. On [2, 3] it is (9 − 4) ÷ 1 = 5.
Step 5: 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?
Falling but Flattening
Problem 2 of 2
Rates over equal intervals are −9, −4, −1. Is the function increasing?
Read the Bend
1 of 8
Rates 2, 4, 6. Concavity?
2 of 8
Rates 6, 3, 0. Concavity?
3 of 8
Rates −6, −3, 0. Concavity?
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?
8 of 8
A cubic changes concavity how many times?
Step 6: Quick Check
Show what you know.
Question 1 of 1
Rates over equal intervals are 1, 5, 11. Concavity?
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.
- Where the rates turn, the concavity changes: an inflection point.