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Precalculus · F.IF.B.4

Concavity: when the rate of change itself changes

Two graphs can both be rising, one faster and faster, the other more and more slowly. How do we say, and measure, which way a rate of change is itself changing?

You need
Rates at an instant from shrinking intervalsPrecalculusIncreasing, decreasing, constant, positive and negative intervalsAlgebra 1
Where you are 10 in Features of a graph
Functions13
What a function is5
Graphs of functions5
Features of a graph10
▴ 5 aboveIncreasing, decreasing, constant, positive and negative intervalsAverage rate of change4
Concavity: when the rate of change itself changes
Continuous graphs, jumps and holesTurning points, maximums and minimums

Why this exists

Increasing and decreasing give the sign of a rate of change, up or down. They say nothing about how the rate itself behaves. A new game whose daily players climb by more each week is in a different situation from one whose players still climb, but by less each week; both graphs are increasing.

Average rates and rates at an instant measure how fast an output changes. Concavity asks the next question: is that rate increasing or decreasing? On a graph the answer shows as the way the curve bends.

Definition

Write \displaystyle m(a, b) = \frac{f(b) - f(a)}{b - a} for the slope of the secant from x = a to x = b.

A function f is concave up on an interval if its secant slopes increase as the secant moves right: whenever a < b < c lie in the interval,

m(a, b) < m(b, c).

It is concave down on the interval if instead m(a, b) > m(b, c) whenever a < b < c lie in it.

When f has a rate of change at each point of the interval, concave up means exactly that this rate increases from left to right, and concave down that it decreases. (Calculus proves that the two forms agree.)

A point of inflection is a point (c, f(c)) on the graph, with f continuous at c, where the concavity changes: concave up on one side of c and concave down on the other.

Intuition

A concave up graph bends like a cup opening upward; a concave down graph bends like a cap. Read left to right, a concave up graph keeps getting steeper in the upward direction: if it is rising, it rises faster; if it is falling, it falls less steeply. A concave down graph is the reverse.

rising faster concave up rising more slowly concave down falling more slowly concave up falling faster concave down
Figure 1Rising or falling says what sign the rate has; concave up or down says whether the rate is increasing or decreasing. All four pairings happen.

So concavity is separate from rising and falling. The sign of the rate says up or down; whether the rate is increasing says which way the graph bends. A point of inflection is where the rate switches between increasing and decreasing, so the rate there is largest or smallest nearby.

Results

Chords. On an interval where f is concave up, the graph between any two of its points lies below the chord joining them. Where f is concave down, it lies above.

Equally spaced tables. If the inputs are equally spaced by d, each secant slope is a first difference divided by d. So on the listed points f is concave up when the first differences increase, that is, when the second differences (the differences of the first differences) are positive, and concave down when they are negative.

Quadratics. The secant slope of f(x) = px^2 + qx + c from a to b is p(a + b) + q. So f is concave up everywhere when p > 0 and concave down everywhere when p < 0, with no point of inflection. With inputs spaced by d, the first differences change by 2pd^2 each step.

Cubics. The rate of change of f(x) = ax^3 + bx^2 + cx + e at x is 3ax^2 + 2bx + c, a quadratic with its vertex at \displaystyle x = -\frac{b}{3a}. So f has exactly one point of inflection, at \displaystyle x = -\frac{b}{3a}. If a > 0 the graph is concave down to its left and concave up to its right; if a < 0, the reverse.

Exponentials. For a > 0, b > 0 and b \ne 1, the first differences of ab^x over steps of equal width increase, whether b > 1 or b < 1: the graph is concave up.

Why it is true

Chords. Take a < x < c in an interval where f is concave up. Since

\begin{aligned} &f(c) - f(a) \\ &\quad = \big(f(x) - f(a)\big) + \big(f(c) - f(x)\big), \end{aligned}

the slope m(a, c) is a weighted average of m(a, x) and m(x, c), with weights \displaystyle \frac{x - a}{c - a} and \displaystyle \frac{c - x}{c - a}. A weighted average lies between the two numbers averaged, and m(a, x) < m(x, c), so m(a, x) < m(a, c). The segment from (a, f(a)) to (x, f(x)) is less steep than the chord, so (x, f(x)) is below the chord.

Quadratics. Factoring,

\begin{aligned} f(b) - f(a) &= p(b^2 - a^2) + q(b - a) \\ &= (b - a)\big(p(a + b) + q\big), \end{aligned}

and dividing by b - a gives m(a, b) = p(a + b) + q. For a < b < c, m(b, c) - m(a, b) = p(c - a), positive exactly when p > 0. Two consecutive steps of width d starting at x have slopes differing by 2pd, so the first differences, d times the slopes, differ by 2pd^2.

Cubics. In Rates at an instant from shrinking intervals the centered average of x^3 at x was 3x^2 + h^2, and that of bx^2 + cx was 2bx + c for every width. Centered averages add, so the centered average of f is 3ax^2 + 2bx + c + ah^2, which settles on 3ax^2 + 2bx + c as h shrinks. That quadratic decreases on one side of its vertex \displaystyle -\frac{2b}{2 \cdot 3a} = -\frac{b}{3a} and increases on the other; with a > 0 it opens upward, so it decreases first.

Exponentials. The first difference over [x, x + d] is ab^{x + d} - ab^x = ab^x(b^d - 1). One step right multiplies it by b^d. If b > 1, it is positive and grows. If b < 1, it is negative and shrinks in size, so it still increases toward 0.

Worked examples

Example 1

Weekly players of a new game

Raffaele tracks the daily players of a new mobile game, averaged over each week, in thousands.

week t0123456
players P2359141718

Describe how the growth changes, and where the concavity changes.

Solution

The weeks are equally spaced, so the first differences are the average rates in thousands of players per week: 1, 2, 4, 5, 3, 1. Their differences, the second differences, are 1, 2, 1, -2, -2.

The players rise every week, so P is increasing throughout. The first differences increase up to the step from week 3 to week 4 and decrease after it: growth speeds up, then slows. So the data are concave up through week 3 and concave down from week 4, where the second differences turn negative. The fastest growth, 5 thousand players a week, came between weeks 3 and 4; a weekly table cannot place the change more closely than that.

Example 2

Secant slopes of a cubic

For f(x) = x^3 - 9x^2 + 24x - 10, find the secant slopes between consecutive integers from x = 0 to x = 6, and use them to describe the concavity.

Solution

The values at x = 0, 1, \dots, 6 are -10, 6, 10, 8, 6, 10, 26; for instance f(3) = 27 - 81 + 72 - 10 = 8. The steps have width 1, so the secant slopes are the first differences:

16, \; 4, \; -2, \; -2, \; 4, \; 16.

The second differences are -12, -6, 0, 6, 12. The slopes decrease on the steps up to x = 3 and increase after it, so the graph looks concave down left of x = 3 and concave up right of it. The two slopes beside x = 3 are equal and the list reads the same in both directions, which suggests the concavity changes exactly at x = 3.

Example 3

The point of inflection, exactly

Prove that the function of Example 2 is concave down on (-\infty, 3) and concave up on (3, \infty), find its point of inflection, and describe how f rises and falls on each stretch.

Solution

By the cubic result, the rate of change of f at x is 3x^2 - 18x + 24. Completing the square,

3x^2 - 18x + 24 = 3(x - 3)^2 - 3.

This rate decreases as x runs up to 3 and increases after, so f is concave down on (-\infty, 3) and concave up on (3, \infty). The point of inflection is (3, f(3)) = (3, 8), where the rate takes its smallest value, -3. The cubic result agrees: \displaystyle -\frac{b}{3a} = -\frac{-9}{3} = 3.

The rate is 3(x - 2)(x - 4), zero at x = 2 and x = 4, where f has its turning points (2, 10) and (4, 6). Combining:

stretchrateconcavityin words
(-\infty, 2)positivedownrising more slowly
(2, 3)negativedownfalling faster
(3, 4)negativeupfalling more slowly
(4, \infty)positiveuprising faster
2 4 6 −10 10 20 30 x y
(3, 8)
 concave down concave up
Figure 2The graph in Examples 2 and 3. Left of x = 3 it bends downward and lies below the dashed tangent line; right of x = 3 it bends upward and lies above it. The point (3, 8) is the point of inflection.

The tangent line at (3, 8) is y = 8 - 3(x - 3) = -3x + 17, and f(x) - (-3x + 17) = (x - 3)^3. That is negative for x < 3 and positive for x > 3: the graph crosses its tangent at the point of inflection, below it on the concave down side and above it on the concave up side.

Example 4

Falling, but more slowly

After an update, Oyunchimeg records the game’s daily players, in thousands, every other day.

day02468
players4031252119

Is the graph concave up or concave down? What does that say about the decline?

Solution

The average rates over each two-day step are first differences divided by 2:

\begin{aligned} \tfrac{31 - 40}{2} &= -4.5, & \tfrac{25 - 31}{2} &= -3, \\ \tfrac{21 - 25}{2} &= -2, & \tfrac{19 - 21}{2} &= -1. \end{aligned}

Every rate is negative, so players are falling. But the rates increase, from -4.5 toward -1 thousand per day, so the data are concave up: the decline is slowing. Concave up does not mean rising.

Example 5

Chords of the square root

1 4 9 1 2 3 x y
\text{slope } 1
\text{slope } \tfrac{1}{3}
\text{slope } \tfrac{1}{5}
Figure 3The chords of Example 5. Moving right, their slopes fall from 1 to \displaystyle \frac{1}{3} to \displaystyle \frac{1}{5}, and the curve bulges above each one.

Show that \displaystyle g(x) = \sqrt{x} is concave down on [0, \infty), starting from its chords on [0, 1], [1, 4] and [4, 9].

Solution

The three slopes are

\frac{1 - 0}{1 - 0} = 1, \qquad \frac{2 - 1}{4 - 1} = \frac{1}{3}, \qquad \frac{3 - 2}{9 - 4} = \frac{1}{5}.

They decrease, which suggests concave down. To prove it for every choice of points, multiply top and bottom by \displaystyle \sqrt{b} + \sqrt{a}, using \displaystyle (\sqrt{b} - \sqrt{a})(\sqrt{b} + \sqrt{a}) = b - a:

m(a, b) = \frac{\sqrt{b} - \sqrt{a}}{b - a} = \frac{1}{\sqrt{a} + \sqrt{b}}.

For 0 \le a < b < c, \displaystyle \sqrt{a} < \sqrt{c}, so \displaystyle \sqrt{a} + \sqrt{b} < \sqrt{b} + \sqrt{c} and

m(a, b) = \frac{1}{\sqrt{a} + \sqrt{b}} > \frac{1}{\sqrt{b} + \sqrt{c}} = m(b, c).

That is the definition of concave down. By the chord result the curve lies above each chord.

Example 6

Growth at a constant percent

Kelechi models the game’s early daily players as N(t) = 4{,}000(1.5)^t after t weeks. Find the first differences over the first four weeks and describe the concavity.

Solution

The values at t = 0, 1, 2, 3, 4 are 4{,}000, 6{,}000, 9{,}000, 13{,}500 and 20{,}250, so the first differences are 2{,}000, 3{,}000, 4{,}500 and 6{,}750.

Each difference is 1.5 times the one before, because N(t + 1) - N(t) = 0.5N(t) and the players keep growing. So the model is concave up for all t, as the exponential result says. Data that bend over, like Raffaele’s, are better fit by logistic models.

Example 7

When were downloads arriving fastest?

Raffaele models the game’s total downloads, in thousands, as D(t) = -t^3 + 12t^2 + 20t for 0 \le t \le 8 weeks. When were downloads arriving fastest, and how fast?

Solution

“Arriving fastest” asks where the rate of D is largest, a question about concavity. By the cubic result, the rate at t is

-3t^2 + 24t + 20 = -3(t - 4)^2 + 68.

This is largest at t = 4, where it is 68 thousand downloads per week. The rate increases on (0, 4) and decreases on (4, 8), so D is concave up, then concave down, with point of inflection (4, D(4)) = (4, 208), since -64 + 192 + 80 = 208.

The total keeps rising after week 4 (at t = 8 the rate is still -3(16) + 68 = 20); the inflection point, not a turning point, marks the busiest week.

Boundaries

  • Concavity belongs to an interval. “Concave up at x = 5” means concave up on some interval around 5.
  • An inflection point must be on the graph. \displaystyle y = \frac{1}{x} is concave down for x < 0 and concave up for x > 0, but x = 0 is not in its domain, so it has no point of inflection.
  • Flat is not inflection. y = x^4 is flat at the origin, yet its secant slopes increase everywhere: it is concave up on the whole line.
  • A line is neither. Its secant slopes are all equal.
  • A table only samples. Differences describe the listed points; between them the function may bend otherwise. Raw differences compare rates only when the inputs are equally spaced.

Where this goes next

In calculus the rate at every input becomes a function of its own, and concavity is read from that function’s rate, the Second and higher derivatives (concavity and inflection points). The S-shaped curves of logistic growth turn on a single point of inflection, where growth is fastest.

Exercises

Tables and differences

  1. The values at x = 0, 1, \dots, 5 are 3, 4, 7, 12, 19, 28. Find the first and second differences and describe the concavity.
  2. Daily players of the game, in thousands, on days 0 to 5 were 50, 59, 66, 71, 74, 75. Describe the growth using first differences.
  3. The values at t = 0, 1, \dots, 6 are 10, 12, 16, 24, 32, 36, 38. Where is the table concave up, where concave down, and near what t does the concavity change?
  4. The values at x = 0, 1, \dots, 5 are 90, 70, 54, 42, 34, 30. Is the table increasing or decreasing? Concave up or down?

Words and sketches

  1. Describe each with “increasing” or “decreasing” and “concave up” or “concave down”. (a) Players are rising, by more each day. (b) Players are falling, by more each day. (c) Players are rising, by less each day. (d) Players are falling, by less each day.
  2. Sketch a graph that is decreasing and concave down on [0, 3] and decreasing and concave up on [3, 6]. What is true of its rate at x = 3?

Secant slopes from a formula

  1. For h(x) = x^3, find the secant slopes on [-2, -1], [-1, 0], [0, 1] and [1, 2], and describe the concavity.
  2. For \displaystyle k(x) = \frac{1}{x}, show that \displaystyle m(a, b) = -\frac{1}{ab} for positive a < b, and find the slopes on [1, 2], [2, 3] and [3, 4]. What is the concavity for x > 0?
  3. For q(x) = 5 - 2x^2, find the slopes on [0, 1], [1, 2] and [2, 3]. By how much does each differ from the last, and why?

Points of inflection of cubics

  1. Find the rate of change of g(x) = x^3 + 6x^2 - 5 at x, the intervals of concavity, and the point of inflection.
  2. Find the intervals of concavity and the point of inflection of p(x) = -2x^3 + 3x^2 + 12x + 1.

The game

  1. Kelechi’s second model is N(t) = 3{,}000(1.2)^t daily players after t days. Find N(0) to N(3) and the first differences. What is the concavity?
  2. After the game’s season ends, daily players are modeled by D(t) = 20{,}000(0.9)^t after t weeks. Find the first differences for t = 0 to 3. Describe the decline.
  3. Oyunchimeg models total downloads, in thousands, as T(t) = -t^3 + 9t^2 + 15t for 0 \le t \le 6 weeks. When were downloads arriving fastest, and how fast? Give the point of inflection.
  4. Raffaele’s readings of daily players, in thousands, were 10 on day 0, 12 on day 1, 18 on day 3, 22 on day 4 and 37 on day 7. Find the average rate over each gap and the concavity of these points. Why do the raw differences mislead?

Reasoning

  1. Explain why a straight line is neither concave up nor concave down.
  2. Explain, with an example from this lesson, why a concave up graph need not be rising.
  3. Show that the secant slope of px^2 + qx + c from a to b is p(a + b) + q, and use it to prove that a quadratic with p < 0 is concave down everywhere.
  4. Kelechi says: “Daily players rose until week 5 and then fell, so week 5 is a point of inflection.” Explain what kind of point week 5 is, and what an inflection point would mean for these players.
  5. Give a function that is increasing and concave down on the whole number line, and explain why it is both.
Answers
  1. First differences 1, 3, 5, 7, 9; second differences all 2. Concave up. The values are x^2 + 3, and 2pd^2 = 2.
  2. First differences 9, 7, 5, 3, 1 thousand a day: rising more slowly, so increasing and concave down. Second differences are all -2.
  3. First differences 2, 4, 8, 8, 4, 2; second differences 2, 4, 0, -4, -2. Concave up from 0 to about 3, concave down after; the change is near t = 3, where the gains are largest.
  4. Decreasing and concave up. First differences -20, -16, -12, -8, -4 increase, with second differences all 4: falling more slowly.
  5. (a) increasing, concave up; (b) decreasing, concave down; (c) increasing, concave down; (d) decreasing, concave up.
  6. The graph falls ever more steeply up to x = 3, then less and less steeply. The rate is negative throughout and most negative near x = 3, a point of inflection.
  7. 7, 1, 1, 7; for example \displaystyle \frac{-1 - (-8)}{1} = 7. Concave down for x < 0, concave up for x > 0, inflection at (0, 0).
  8. \displaystyle \frac{1/b - 1/a}{b - a} = \frac{a - b}{ab(b - a)} = -\frac{1}{ab}. The slopes are \displaystyle -\frac{1}{2}, \displaystyle -\frac{1}{6}, \displaystyle -\frac{1}{12}, increasing, so k is concave up for x > 0.
  9. -2, -6, -10, from q = 5, 3, -3, -13. Each is 4 less: 2pd^2 = 2(-2)(1)^2 = -4.
  10. Rate 3x^2 + 12x = 3(x + 2)^2 - 12. Concave down on (-\infty, -2), concave up on (-2, \infty); inflection point (-2, 11), since g(-2) = -8 + 24 - 5 = 11.
  11. Rate -6x^2 + 6x + 12, with vertex at \displaystyle x = \frac{1}{2}. Concave up on \displaystyle \left(-\infty, \frac{1}{2}\right), concave down on \displaystyle \left(\frac{1}{2}, \infty\right); inflection point \displaystyle \left(\frac{1}{2}, \frac{15}{2}\right), since \displaystyle -\frac{1}{4} + \frac{3}{4} + 6 + 1 = \frac{15}{2}.
  12. 3{,}000, 3{,}600, 4{,}320, 5{,}184; differences 600, 720, 864, each 1.2 times the last. Concave up.
  13. Values 20{,}000, 18{,}000, 16{,}200, 14{,}580; differences -2{,}000, -1{,}800, -1{,}620. Decreasing and concave up: players fall by less each week.
  14. Rate -3t^2 + 18t + 15 = -3(t - 3)^2 + 42, largest at t = 3: 42 thousand downloads per week. Inflection point (3, 99), since -27 + 81 + 45 = 99.
  15. Rates 2, 3, 4, 5 thousand per day, from \displaystyle \frac{2}{1}, \displaystyle \frac{6}{2}, \displaystyle \frac{4}{1}, \displaystyle \frac{15}{3}. They increase: concave up. The raw differences 2, 6, 4, 15 cover gaps of different lengths, so they are not rates.
  16. Every secant of y = mx + c has slope m, so m(a, b) = m(b, c), never less and never greater.
  17. In Example 4 players fall every day, yet the rates -4.5, -3, -2, -1 increase: concave up. Concave up says the rate increases, not that it is positive.
  18. Factor f(b) - f(a) as (b - a)\big(p(a + b) + q\big) and divide by b - a. For a < b < c, m(b, c) - m(a, b) = p(c - a) < 0, so the slopes decrease: concave down.
  19. Week 5 is a turning point, a relative maximum, where the rate changes sign. An inflection point is where the rate switches between increasing and decreasing: here, if there was one, the week growth was fastest before week 5, or decline fastest after it.
  20. For example y = -(0.5)^x. Its first differences over width d are (0.5)^x\left(1 - (0.5)^d\right): positive, so increasing, and each (0.5)^d times the last, so shrinking, and concave down.
kg1128 · updated 2026-10-20 · CC-BY-SA-4.0