Concavity Describes How a Graph Bends
A function can rise while its graph bends upward, or rise while its graph bends downward. It can also fall while bending in either of those ways. To describe the bending, we compare slopes across an interval rather than asking only whether the function’s values increase or decrease.
The slope of a secant line records the average change in function value between two inputs. If secant slopes become larger as we move from left to right, the graph bends upward in the sense of concavity. If they become smaller, it bends downward. This comparison gives a precise definition that works directly with a graph or with values at selected points.
Secant Slopes and Concavity
For a function \(f\) and inputs \(u<v\) in its domain, the secant slope from \(u\) to \(v\) is the slope of the line through \((u,f(u))\) and \((v,f(v))\):
The denominator is positive because \(u<v\). To compare the graph’s bending, choose three inputs \(a<b<c\). Compare the secant slope over the left interval \([a,b]\) with the one over the right interval \([b,c]\).
The inequalities must hold for every triple of ordered inputs to establish concavity on the whole interval. A comparison for one triple tells us about those sampled intervals only. It does not, by itself, establish how the function behaves everywhere between or beyond the points.
Why the Slope Definition Matches the Chord Picture
For three points \(a<b<c\), the chord from \((a,f(a))\) to \((c,f(c))\) has a particular height at \(x=b\). That height is the weighted average of the endpoint values with weights determined by the distances. The slope definition says exactly whether \(f(b)\) lies below or above that chord.
Here is the algebra that proves the claim. Since \(b-a\) and \(c-b\) are positive, multiplying by their product preserves the inequality:
In the last step we divided by \(c-a>0\). The expression on the right is indeed the chord height: moving from \(a\) to \(b\) covers the fraction \((b-a)/(c-a)\) of the full horizontal distance, so the chord’s value there is \(f(a)+\frac{b-a}{c-a}(f(c)-f(a))\). Expanding this gives the stated weighted average. Thus, for a concave-up function, every point of the graph between two endpoints lies on or below their chord. For a concave-down function, it lies on or above the chord.
Worked Example: Classifying Concavity from Three Points
Worked Example: Classifying Concavity from Three Points
Suppose a graph passes through \((-1,3)\), \((1,1)\), and \((4,10)\). Compare the two secant slopes to determine how these three points are ordered in concavity.
The left slope, from \(x=-1\) to \(x=1\), is
The right slope, from \(x=1\) to \(x=4\), is
Since \(-1\leq 3\), the secant slope increases across these two intervals. The three points satisfy the concave-up slope comparison. Check the chord-height form: at \(x=1\), the chord from \(x=-1\) to \(x=4\) has height
The actual value \(f(1)=1\) satisfies \(1\leq 29/5\), as expected. This verifies the comparison for the given points; it does not prove concavity of an entire function unless the slope condition is known for every triple in its interval.
Worked Example: Establishing Concavity Up for a Function
Worked Example: Establishing Concavity Up for a Function
Consider \(f(x)=x^2\) on \(\mathbb{R}\). For any \(a<b<c\), first calculate each secant slope:
The difference between the right slope and the left slope is
Therefore \(m_{a,b}<m_{b,c}\) for every \(a<b<c\), so \(x^2\) is concave up on \(\mathbb{R}\). As a numerical check, use \(a=-1\), \(b=1\), and \(c=3\). The function values are \(1\), \(1\), and \(9\), giving slopes \((1-1)/(1-(-1))=0/2=0\) and \((9-1)/(3-1)=8/2=4\). Their difference is \(4-0=4\), which also equals \(c-a=3-(-1)=4\).
Worked Example: Reading a Piecewise-Linear Graph
Worked Example: Reading a Piecewise-Linear Graph
Suppose a graph consists of straight segments joining, in order, the points \((-2,2)\), \((0,0)\), \((2,2)\), and \((4,8)\). Find the slopes of its three successive segments.
From \((-2,2)\) to \((0,0)\), the slope is
From \((0,0)\) to \((2,2)\), it is
From \((2,2)\) to \((4,8)\), it is
These segment slopes are \(-1\), \(1\), and \(3\), so they increase from left to right. Within each straight segment the slope is constant, and across the joins it never decreases. Consequently, for any two successive intervals, their secant slopes are nondecreasing: each secant slope is an average of the segment slopes over the interval it covers, and every segment slope farther to the right is at least as large as every segment slope farther to the left. Thus the whole piecewise-linear graph is concave up. As a check on the first and last listed segments, their slopes satisfy \(-1\leq 3\); the middle slope \(1\) lies between them.
Concavity Is Not the Same as Increasing or Decreasing
A secant slope can be negative, zero, or positive. Concavity concerns how such slopes compare as the input moves from left to right; it is not determined by the sign of a single slope. For example, the first slope in the piecewise-linear example is negative, but the successive slopes still increase. That portion of the graph is falling while bending upward.
Likewise, a function can rise while bending downward. If the slopes of successive chords are \(5\) and then \(2\), both are positive, so the graph rises over both intervals; because the slope has decreased, the three-point comparison indicates concave-down behavior.
A Common Pitfall: Classifying from a Slope’s Sign
Suppose the secant slopes over successive intervals are \(-4\) and \(-2\). A mistaken argument is: “Both slopes are negative, so the graph is concave down.” That conclusion confuses the sign of the slopes with their order. The correct comparison is
The slopes have increased, so the points satisfy the concave-up comparison. The graph is falling over both intervals, but it is falling less steeply on the second interval. To decide concavity, compare the slopes themselves, not whether they are positive or negative.
Another important caution is that a few plotted points cannot establish a global claim about an otherwise unspecified graph. A drawn curve might bend between the samples in a way that changes the slope comparisons. Use finite data to classify the stated triples or intervals; to claim concavity on an entire interval, the defining comparison must hold for every ordered triple there.
Using the Definition in Practice
Use \(a<b<c\), so the left and right intervals are unambiguous.
For point data, use the change in output divided by the change in input on each interval.
If the left slope is no greater than the right slope, the triple has the concave-up ordering; if it is no smaller, it has the concave-down ordering.
A comparison classifies the chosen triple. A whole-interval claim requires the appropriate inequality for every triple in that interval.
On a graph, the same method can be used qualitatively. Compare chord slopes across successive intervals: “larger” means more positive or less negative, and “smaller” means less positive or more negative. If the graph is given only as a sketch, report the conclusion as a reading of that graph rather than as a proof about an exact function.
Check Your Understanding
Use the secant-slope definition to answer these questions.
- For three points with successive secant slopes \(2\) and \(5\), which concavity comparison do they satisfy?
- A graph has successive secant slopes \(-3\) and \(-1\). Is that pattern consistent with a graph that is falling but concave up?
- For \(f(x)=x^2\), what are the secant slopes over \([-2,0]\) and \([0,2]\)? What does their comparison show for those intervals?
- Why does observing three points with the concave-up slope ordering not, by itself, prove that a function is concave up on its entire domain?
- For \(a<b<c\), how does the concave-up slope inequality determine whether \(f(b)\) lies above or below the chord joining the endpoint values?
