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: Picture It
Start with something you already know.
A staircase with uneven steps
- Picture a staircase where every step is the same width.
- Only the height of each step can change.
- If each step is taller than the last, the climb gets steeper as you go.
- If each step is shorter than the last, you still climb, but the stairs flatten out.
- Going downstairs, drops that shrink mean you are still going down, but more and more gently.
Each step is one equal interval, and its height is the average rate of change over it.
Rises that grow make a bowl, concave up; rises that shrink make a dome, concave down.
Downstairs, drops that shrink are rates rising toward zero: concave up, though you are going down.
The same idea with numbers: three steps, each one stride wide.
| Step | How much it rises |
|---|---|
| First | 20 cm |
| Second | 15 cm |
| Third | 10 cm |
Each rise is smaller than the last: still climbing, but flattening, like a dome. That is concave down.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
A toy cart rolls down a ramp. Its distance from the top is 0 m at 0 seconds, 2 m at 1 second, 8 m at 2 seconds, and 18 m at 3 seconds. Find its average speed over each one-second stretch. What do those speeds do?
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
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 3: 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 4: 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 5: Watch an Example
One step at a time.
Watch Noor Test a Dome
A fresh function: f(x) = 6x − x². Noor finds its average rate of change on [0, 1], [1, 2] and [2, 3], and reads its concavity from them.
- Step 1
Noor first finds the function at each whole number from 0 to 3. f(0) = 0, f(1) = 6 − 1 = 5, f(2) = 12 − 4 = 8, and f(3) = 18 − 9 = 9.
Watch Leo Start It, Then Finish It Yourself
f(x) = x² − 8x on [0, 4], in unit intervals. Leo finds the values and the first two rates; the rest is yours.
- Step 1
Leo works out the five values first: f(0) = 0, f(1) = 1 − 8 = −7, f(2) = 4 − 16 = −12, f(3) = 9 − 24 = −15, and f(4) = 16 − 32 = −16.
Watch Ravi Find Where the Bend Turns
f(x) = x³ − 3x² on [−1, 3], in unit intervals. Ravi looks for the place where the concavity changes.
- Step 1
Ravi finds the values: f(−1) = −1 − 3 = −4, f(0) = 0, f(1) = 1 − 3 = −2, f(2) = 8 − 12 = −4, and f(3) = 27 − 27 = 0.
Step 6: Your Turn
Practice makes it stick.
The Growing Gains
Problem 1 of 14
A savings account holds $500 at the start. At the end of the first three months it holds $520, $545 and $575. Which describes the balance over those three months?
The Braking Car
Problem 2 of 14
A car brakes gently. It travels 18 m in the first second, 14 m in the next, and 10 m in the one after. Its speed keeps dropping by the same amount each second. How many meters does it travel in the fourth second?
meters
Read the Bend
Find the rates over equal intervals first. Then ask what the rates are doing.
Problem 3 of 14
f(x) = 3x². What is its average rate of change on [2, 3]?
Problem 4 of 14
A function has f(0) = 0, f(1) = 12, f(2) = 19 and f(3) = 22. Which way is it concave on [0, 3]?
Problem 5 of 14
f(x) = 2x² − x. Using [0, 1], [1, 2] and [2, 3], what is its second difference?
Problem 6 of 14
A function has f(0) = 0, f(1) = 7, f(2) = 8, f(3) = 9 and f(4) = 16. At which x does its concavity change?
Whose Step?
One line on each page is a slip. Find it, then put it right.
Problem 7 of 14
Ana tested the concavity of f(x) = x² + 2x on [0, 3]. One line is a slip. Which?
Ana’s working for the concavity of f(x) = x² + 2x
- 1f(0) = 0, f(1) = 3, f(2) = 8, f(3) = 15
- 2[0, 1]: (3 − 0)/1 = 3
- 3[1, 2]: (8 − 0)/1 = 8
- 4[2, 3]: (15 − 8)/1 = 7
- Answer3, 8, 7: the rates rise then fall, so the concavity changes
Problem 8 of 14
What is the average rate of change of f(x) = x² + 2x on [1, 2], correctly?
Problem 9 of 14
Maya described a function from four of its values. One line is a slip. Which?
Maya’s working for f(0) = 20, f(1) = 12, f(2) = 7, f(3) = 5
- 1f(0) = 20, f(1) = 12, f(2) = 7, f(3) = 5
- 2rates: −8, −5, −2
- 3every rate is negative: decreasing
- Answerdecreasing, so concave down
Problem 10 of 14
Which way is Maya’s function concave on [0, 3], correctly?
Pick the Move
Decide what to work out. Neither question needs a number.
Problem 11 of 14
You know f(0), f(1), f(2) and f(3) for a function. To decide its concavity on [0, 3], what do you work out first?
Problem 12 of 14
You have a function’s rates over equal intervals. You want to know whether it is increasing, and which way it is concave. What do you look at?
Same Bend?
Judge each one by what its rates do. Direction does not decide it.
Problem 13 of 14
Each list is a function’s rates over equal intervals. Select every list that shows the same bend as x², concave up.
Problem 14 of 14
Each list is a function’s values at x = 0, 1, 2, 3. Select every one that is concave down, a dome, on [0, 3].
Step 7: Quick Check
Show what you know.
Question 1 of 1
Rates over equal intervals are 1, 5, 11. Concavity?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
A function has f(0) = 0, f(1) = 10 and f(2) = k. It must rise on [1, 2], and its rates on [0, 1] and [1, 2] must show concave down. How many whole numbers could k be?
Stretch 2 of 2
A quadratic has f(0) = 3, f(1) = 5 and f(2) = 9. What is f(3)?
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.
For the grown-up
The misconception this lesson targets is tying concavity to direction: reading "going down" as "concave down" and "going up" as "concave up". Direction comes from the signs of the rates; concavity comes from whether the rates rise or fall, and the two are independent. Maya’s page in Whose Step? makes exactly this slip, and the braking car and Leo’s falling-but-flattening curve show the other two mixes. Its cousin, named in the theory, is reading the bend from whether the graph sits above or below the axis. Any doubt is settled the same way: list the rates over equal intervals and compare each one with the one before.