Concavity and Points of Inflection
Picture a roller coaster. At the bottom of a dip, the track bends upward like a bowl. At the top of a hill, it bends downward like an arch. Somewhere between a dip and the next hill, the bending switches. Where is that spot, and what does it tell us about the ride? By the end of this page you will be able to answer that for any graph.
Concave up and concave down
A curve can bend in two ways.
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Concave up ($\cup$): the graph bends like a bowl or a smile. It could "hold water."
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Concave down ($\cap$): the graph bends like an arch or a frown. Water would run off it.
Some problems call a concave-up graph convex and a concave-down graph concave. They are the same two ideas with different labels, so check the symbol $\cup$ or $\cap$ if you are unsure which one a problem means.
There is a second way to see it, and it is the one that matters most in applications. Look at the slope as you move from left to right.
The slopes $1, 3, 5, 7$ keep increasing. That is what concave up means: the graph gets steeper (or less steeply downward) as $x$ increases. For a concave-down graph the slopes keep decreasing.
Explore a parabola with the slider. Watch what happens when $a$ is positive, negative, and exactly $0$.
A skating ramp (half-pipe type) has a cross section whose shape can be modeled mathematically by the parabola $y = 0.5x^2$. What type of curvature does the ramp present,…
A skating ramp (half-pipe type) has a cross section whose shape can be modeled mathematically by the parabola $y = 0.5x^2$.
What type of curvature does the ramp present, concave ($\cap$) or convex ($\cup$)? Is there any inflection point in this continuous structure?
During a basketball game, a player throws the ball toward the basket. The trajectory of the ball follows the equation $h(t) = -5t^2 + 6t + 2$, where $h$ is the height in…
During a basketball game, a player throws the ball toward the basket. The trajectory of the ball follows the equation $h(t) = -5t^2 + 6t + 2$, where $h$ is the height in feet and $t$ is the time in seconds.
Is the trajectory described by the ball concave ($\cap$) or convex ($\cup$)? Is there any inflection point in the flight of the ball? Reason your answer based on the shape of the graph.
Parabolas: the leading coefficient decides
For a quadratic function $f(x) = ax^2 + bx + c$, you do not need a graph to know the bending. The sign of the leading coefficient $a$ tells you everything.
Why does this work? The slope of a parabola changes at a steady rate. When $a>0$ the slope always increases, and when $a<0$ it always decreases. It never switches from one to the other.
When the coefficient has a parameter
Sometimes the coefficient contains a letter. Then you set up an inequality.
Classify the following quadratic functions as concave ($\cap$) or convex ($\cup$) without graphing them. Remember to look at the leading coefficient. a)…
Classify the following quadratic functions as concave ($\cap$) or convex ($\cup$) without graphing them. Remember to look at the leading coefficient.
- a) $f(x) = 3x^2 - 2x + 1$
- b) $g(x) = -x^2 + 5x$
- c) $h(x) = \frac{1}{2}x^2 - 4$
- d) $i(x) = 8 - 2x^2$
Determine the values of the parameter $k$ for which the quadratic function $f(x) = (2k - 8)x^2 + 3x - 5$ is a concave parabola ($\cap$). For what exact value of $k$…
Determine the values of the parameter $k$ for which the quadratic function $f(x) = (2k - 8)x^2 + 3x - 5$ is a concave parabola ($\cap$).
For what exact value of $k$ would the function cease to have curvature? What type of function would be obtained in that specific case?
Consider the family of parametric quadratic functions given by the expression $f(x) = (a^2 - 9)x^2 + 5x - 2$, where $a$ is a real number. a) For what exact values of the…
Consider the family of parametric quadratic functions given by the expression $f(x) = (a^2 - 9)x^2 + 5x - 2$, where $a$ is a real number.
- a) For what exact values of the parameter $a$ is the function a convex parabola ($\cup$)?
- b) For what values of $a$ is it a concave parabola ($\cap$)?
- c) Is there any value of $a$ for which the graph of the function lacks curvature? In that case, what type of geometric function is obtained?
Adding two functions
What happens to the bending when you add two functions? Think about slopes. The slope of a sum is the sum of the slopes. If both slopes keep increasing, their sum keeps increasing too.
Be careful: this does not work for a concave-up function plus a concave-down one. There the result depends on which one "wins."
For quadratics you can see it directly in the formula.
Prove whether the sum of two convex functions ($\cup$) always results in another convex function. To investigate this algebraically, take the parabolas $f(x) = x^2$ and…
Prove whether the sum of two convex functions ($\cup$) always results in another convex function.
To investigate this algebraically, take the parabolas $f(x) = x^2$ and $g(x) = 2x^2 - 4x + 1$, both convex. Find the expression for the sum function $h(x) = f(x) + g(x)$ and analyze the type of curvature that $h(x)$ presents.
Lines and other basic curves
Not every function bends. And the ones that do can bend the same way forever.
| Function | Curvature |
|---|---|
| Line $y = mx + b$ | None. It is straight, so there is no concave up or down and no inflection point. |
| Parabola $y = ax^2 + bx + c$ | Always concave up ($a>0$) or always concave down ($a<0$). |
| Exponential growth $y = ab^x$ with $a > 0$, $b > 1$ | Always concave up. |
| Inverse variation $y = k/x$ with $k > 0$ | Concave up for $x>0$, concave down for $x<0$. |
A line has a constant slope, so the slope never increases or decreases. That is why a total cost like $C(x) = 5x + 1000$ has no curvature at all.
For $y = \dfrac{k}{x}$ with $k>0$, the graph has two separate branches. On the right branch $(0, \infty)$ it bends like a bowl. On the left branch $(-\infty, 0)$ it bends like an arch. Be careful about whether there is an inflection point at $x=0$ (see the next section: the function must be defined there).
For exponential growth, the slope keeps getting bigger. Each year a savings account earns more dollars than the year before, so the graph never flattens out and never switches to concave down.
Given the inverse proportion function $f(x) = \frac{3}{x}$, analyze its graph mentally or by drawing a quick sketch. What type of curvature does it have in the interval…
Given the inverse proportion function $f(x) = \frac{3}{x}$, analyze its graph mentally or by drawing a quick sketch.
What type of curvature does it have in the interval $(0, +\infty)$? And in the interval $(-\infty, 0)$? Can it be said that there is an inflection point at $x=0$?
A 3D printing company has fixed monthly costs of $\$1000$ and a production cost of $\$5$ for each part manufactured. The total cost function is given by…
A 3D printing company has fixed monthly costs of $\$1000$ and a production cost of $\$5$ for each part manufactured. The total cost function is given by $C(x) = 5x + 1000$.
Does this function have concavity, convexity, or inflection points? Justify your answer based on the graphical representation of the function.
An initial capital of $\$1000$ is invested in a fund at an annual compound interest of $5\%$. The function that describes the accumulated capital over time $t…
An initial capital of $\$1000$ is invested in a fund at an annual compound interest of $5\%$. The function that describes the accumulated capital over time $t$ (in years) is $C(t) = 1000 \cdot (1.05)^t$.
Knowing that exponential functions with a base greater than $1$ have a characteristic curvature, determine whether this function is concave ($\cap$) or convex ($\cup$). Will there come a time in the future when the rate of capital growth begins to decrease, thus generating an inflection point? Justify your answer.
Inflection points
Now we can answer the roller-coaster question.
Here is the simplest example, $y = x^3$.
To the left of the origin the curve bends like an arch. To the right it bends like a bowl. At $(0, 0)$ the switch happens, so the origin is the inflection point. The tangent line at an inflection point crosses through the curve from one side to the other.
Inflection points and turning points are different
A turning point (local maximum or minimum) is where the graph switches between rising and falling. An inflection point is where the graph switches between the two bendings. They are different ideas. For $y=x^3$, the graph rises everywhere, yet it has an inflection point.
Peaks and valleys
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A smooth local maximum (the top of a hill) can only occur in a concave-down region.
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A smooth local minimum (the bottom of a valley) can only occur in a concave-up region.
Think of the hilltop and the valley floor: the hill is an arch and the valley is a bowl. So between a peak and the next valley, the graph must switch bendings. There is an inflection point in between.
Cubics
A cubic with two turning points has exactly one inflection point, and it sits halfway between them. In fact, the cubic is symmetric about that point.
Look at the following graph of a polynomial function of degree 3. State in which interval the function is concave ($\cap$) and in which it is convex ($\cup$). Identify…
Look at the following graph of a polynomial function of degree 3.
State in which interval the function is concave ($\cap$) and in which it is convex ($\cup$). Identify the exact coordinates of the point $P$, knowing that it is the inflection point and is located at the origin.
State whether the following statement is true or false and justify your answer in detail: "An inflection point is that point on the graph where the function changes from…
State whether the following statement is true or false and justify your answer in detail:
"An inflection point is that point on the graph where the function changes from being increasing to being decreasing, or vice versa."
Given the polynomial function $f(x) = x^3 - 6x^2 + 9x$, it is known from its graph that its inflection point is located exactly at the coordinate $x$ which is the…
Given the polynomial function $f(x) = x^3 - 6x^2 + 9x$, it is known from its graph that its inflection point is located exactly at the coordinate $x$ which is the midpoint between its two relative extrema (a local maximum at $x=1$ and a local minimum at $x=3$).
Calculate the exact coordinates of the inflection point and determine the intervals of concavity ($\cap$) and convexity ($\cup$) of the function.
Corners and joins
Two special situations show up often: graphs with sharp corners and graphs built from several pieces.
Corners
An inflection point needs a smooth graph, one where you could draw a single tangent line. A sharp corner has no single tangent line. Take $f(x) = |x^2 - 4|$.
The graph is concave up on $(-\infty, -2)$, concave down on $(-2, 2)$, and concave up on $(2, \infty)$. The concavity does change at $x = \pm 2$. But the graph arrives at a sharp point there, with no single tangent line. For that reason we do not call these corner points proper inflection points. They are corners where the bending switches.
Pieces joined together
A piecewise function can also change concavity at the place where its pieces meet. To decide, do three checks.
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Is the graph connected there? Compare the values from both pieces.
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What is the concavity of each piece? Use the leading coefficient.
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Does the graph join smoothly (no corner), and does the concavity switch?
If the answer is yes to all three, you have an inflection point at the junction.
The function $f(x) = |x^2 - 4|$ represents the absolute value of a parabola and its graph has a rounded "W" shape. Look at its graph and determine in which intervals the…
The function $f(x) = |x^2 - 4|$ represents the absolute value of a parabola and its graph has a rounded "W" shape.
Look at its graph and determine in which intervals the function is concave ($\cap$) and in which it is convex ($\cup$). Can the points $x=-2$ and $x=2$ (marked in red) be considered proper inflection points? Reason your answer.
The following graph corresponds to a piecewise function, composed of two fragments of different parabolas that join smoothly. Determine the intervals of concavity and…
The following graph corresponds to a piecewise function, composed of two fragments of different parabolas that join smoothly.
Determine the intervals of concavity and convexity of the global function. What does the orange junction point at $x=0$ represent in relation to the curvature of the function?
Given the following piecewise quadratic function: $f(x) = \begin{cases} -x^2 + 4x & \text{if } x < 2 \\ x^2 - 4x + 8 & \text{if } x \ge 2 \end{cases}$ a) Prove…
Given the following piecewise quadratic function:
$f(x) = \begin{cases} -x^2 + 4x & \text{if } x < 2 \\ x^2 - 4x + 8 & \text{if } x \ge 2 \end{cases}$
- a) Prove algebraically whether the function is continuous at the breakpoint $x = 2$.
- b) Analytically study the curvature of each of the two sections.
- c) Can it be rigorously stated that there is an inflection point at $x = 2$? Reason your answer based on the definition of an inflection point.
Curvature and rates of change
Here is the idea that makes concavity useful in real life: the slope of a graph is a rate of change, so the concavity tells you whether that rate is increasing or decreasing.
Some examples:
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Distance-time graph: the slope is speed. Concave up means speeding up. Concave down means slowing down. An inflection point is the moment you switch.
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Total (cumulative) graph, such as total infections or total views: the slope is "new per day." Concave up means new cases per day are growing. Concave down means they are shrinking. The inflection point is when new daily cases are at their peak.
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Altitude-time graph of a drone: concave up means the climb is getting faster, and concave down means it is getting slower.
You can spot concavity in a table by looking at the change over equal steps.
Here is a graph that has both bendings: a car speeds up, then slows down.
The graph is concave up until $t = 4$ (speeding up), then concave down (slowing down). The inflection point $(4, 64)$ is the moment of greatest speed.
From a graph of the rate
Sometimes you are given the graph of the rate itself. If the rate graph is decreasing (like the line $y = -2x + 4$), the rate is always decreasing, so the original function is concave down everywhere. Notice that the rate can still be positive: a decreasing rate only says the function climbs more and more slowly.
The following graph represents the altitude of a surveillance drone over time during an ascent maneuver. Identify what type of curvature the graph has before and after…
The following graph represents the altitude of a surveillance drone over time during an ascent maneuver.
Identify what type of curvature the graph has before and after the point $I$. Explain what this inflection point means physically in the context of the drone's flight (how its way of ascending changes).
The following graph shows the total accumulated number of people infected by a virus in a population over the days of a month. a) Visually identify the intervals where…
The following graph shows the total accumulated number of people infected by a virus in a population over the days of a month.
- a) Visually identify the intervals where the infection curve is convex ($\cup$) and where it is concave ($\cap$).
- b) Locate the inflection point (marked in orange). What crucial meaning does this point have in the evolution of the epidemic with respect to the number of new daily infections?
Suppose that the rate of change* (the growth rate or gradient) of a function $f(x)$ over time is given by the line $y = -2x + 4$. Knowing that the growth rate of $f(x)$…
Suppose that the rate of change (the growth rate or gradient) of a function $f(x)$ over time is given by the line $y = -2x + 4$.
Knowing that the growth rate of $f(x)$ is decreasing (since the orange line representing the rate is strictly decreasing), what can you deduce about the overall curvature of the original function $f(x)$? Will the original function have any inflection point? Explain your answer.
Filling containers and journeys
If you pour water into a container at a constant rate, the water's height changes depending on how wide the container is at that level.
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Wide spot: the same volume spreads over a large area, so the height rises slowly.
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Narrow spot: the same volume stacks up high, so the height rises quickly.
So as the container gets wider, the rise of the water slows down: the slope decreases, and the graph is concave down. As the container gets narrower, the rise speeds up: the slope increases, and the graph is concave up. Straight vertical sides give a straight line.
A journey works the same way: it can be broken into stretches where the speed increases, stays constant, or decreases. Each stretch gives a different concavity on the distance graph. An inflection point appears when the stretch changes from "speeding up" to "slowing down" (or the reverse).
Imagine that you are filling with water, at a constant rate, a glass bottle that has a wide spherical base and a narrow cylindrical neck. If we graphically represent the…
Imagine that you are filling with water, at a constant rate, a glass bottle that has a wide spherical base and a narrow cylindrical neck.
If we graphically represent the height of the water in the bottle in terms of the elapsed time, describe what the curvature of the graph is like in the lower spherical zone (where the bottle widens and then narrows). Is there any inflection point? Where would it be located approximately?
A high-speed train makes a journey in a straight line. During the first $4$ minutes, it accelerates with increasing intensity. Then, from minute $4$ to $8$, it continues…
A high-speed train makes a journey in a straight line. During the first $4$ minutes, it accelerates with increasing intensity. Then, from minute $4$ to $8$, it continues to accelerate but at a decreasing rate until it reaches its maximum cruising speed. Finally, from minute $8$ to $12$, it brakes progressively until it stops at the next station.
If we represent the distance traveled in terms of time, what are the intervals of concavity and convexity of this position graph? Identify the inflection points and explain their physical meaning in relation to the train's acceleration.
A water tank has the shape of an inverted cone (the point downwards and the circular base open upwards). Water begins to be poured into it at a constant rate of gallons…
A water tank has the shape of an inverted cone (the point downwards and the circular base open upwards). Water begins to be poured into it at a constant rate of gallons per minute.
If we construct a graph that relates the height of the water level $h(t)$ in terms of the elapsed time $t$, will the resulting curve be concave ($\cap$) or convex ($\cup$)? Justify your answer by analyzing how the speed at which the water level rises varies as the cone widens.
Putting it all together
Now combine everything: symmetry, peaks and valleys, and concavity.
Odd symmetry. If a graph is symmetric about the origin, then the concavity on one side of the origin is the opposite of the concavity on the other side. A concave-up piece on the right corresponds to a concave-down piece on the left, and vice versa.
Peaks and valleys. Remember that smooth peaks sit in concave-down regions and smooth valleys sit in concave-up regions.
Leveling off. If a curve rises to a peak, then falls and levels off toward a horizontal line, the tail must bend back upward (the slope goes from steeply negative back toward $0$). So an inflection point sits between the peak and the tail.
Here is a rational function with the "leveling off" shape. Look for the three places where the bending switches.
Here the inflection points are at $x=0$ and, on either side, one between the peak and the tail. By symmetry they are mirror images through the origin.
The profile of a mountain cycling stage can be modeled in a specific section by the function $h(x) = -0.1x^3 + 1.5x^2 + 2x + 500$, where $x$ is the horizontal distance…
The profile of a mountain cycling stage can be modeled in a specific section by the function $h(x) = -0.1x^3 + 1.5x^2 + 2x + 500$, where $x$ is the horizontal distance in miles and $h(x)$ is the altitude in feet. It is known that exactly at mile $x=5$ the upward slope stops increasing and begins to decrease progressively until reaching the top.
What name is given to the point $x=5$ in terms of function analysis? What type of curvature does the mountain profile have before and after that mile?
The attached graph corresponds to the rational function $f(x) = \frac{100x}{x^2+4}$. From a detailed observation of the curve and the marked orange points, identify how…
The attached graph corresponds to the rational function $f(x) = \frac{100x}{x^2+4}$.
From a detailed observation of the curve and the marked orange points, identify how many inflection points the function has and determine approximately the intervals of concavity ($\cap$) and convexity ($\cup$) in its entire domain $\mathbb{R}$.
Describe and draw a sketch in your notebook of a continuous function $f(x)$ defined in the domain $[-5, 5]$ that simultaneously satisfies all the following conditions:…
Describe and draw a sketch in your notebook of a continuous function $f(x)$ defined in the domain $[-5, 5]$ that simultaneously satisfies all the following conditions:
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It has odd symmetry (symmetry with respect to the origin).
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It has a relative maximum at $x = 2$ and a relative minimum at $x = -2$.
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It has exactly three inflection points located at $x = -3$, $x = 0$ and $x = 3$.
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It is convex ($\cup$) in the intervals $[-5, -3) \cup (0, 3)$.
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It is concave ($\cap$) in the intervals $(-3, 0) \cup (3, 5]$.
Summary
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Concave up ($\cup$): bends like a bowl, and the slope is increasing. Concave down ($\cap$): bends like an arch, and the slope is decreasing.
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For $ax^2 + bx + c$: $a>0$ is concave up, $a<0$ is concave down, and $a=0$ is a line with no curvature. A parabola has no inflection point.
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The sum of two concave-up functions is concave up. For quadratics, add the leading coefficients to check.
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An inflection point is a smooth point where the concavity changes. It is not the same as a turning point (local max or min).
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Smooth peaks sit in concave-down regions, and smooth valleys in concave-up regions. A cubic's inflection point is halfway between its turning points.
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Sharp corners are not proper inflection points. For piecewise functions, check continuity, concavity on each side, and smoothness.
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Concavity tells you if a rate is increasing or decreasing: speeding up or slowing down, new daily cases growing or shrinking, a container widening or narrowing.
Back to the roller coaster: the spot where the track switches from a dip to a hill is an inflection point. There the slope of the track stops getting steeper and starts to ease off.