Determine intervals of concavity from a graph

From a graph, determine intervals of concavity by interpreting how the function’s slopes change from left to right: concave up where slopes increase and concave down where slopes decrease. Partition the domain at points where the graph changes concavity, identifying such points as possible inflection points; do not confuse increasing or decreasing function values, or a local maximum or minimum, with concavity. This graphical analysis is limited to the displayed domain and supports later interpretation of second-derivative information.

Detailed Explanation: Determine intervals of concavity from a graph

Look at how the slopes of the graph change as you move from left to right.

Concave up: slopes are increasing. For example, slopes might change from $-4$ to $-2$ to $0$ to $3$ .
Concave down: slopes are decreasing. For example, slopes might change from $3$ to $1$ to $-1$ .

This is about how the graph bends, not whether the function values are increasing or decreasing.

Worked example

Suppose a graph is shown on the domain $[-4,4]$ . By examining the graph, you estimate the slopes at several points:

\begin{array}{c|ccccc} x & -3 & -1 & 0 & 1 & 3\\ \hline \text{slope} & -4 & -2 & -1 & 1 & -2 \end{array}

Step 1: Examine the slopes before $x=1$ .

From $x=-3$ to $x=1$ , the slopes change like this:

-4,\ -2,\ -1,\ 1

The slopes are increasing, so the graph is concave up on

(-4,1).

Notice that some slopes are negative. The graph may still be decreasing on part of this interval, but it is concave up because the slopes are becoming larger.

Step 2: Examine the slopes after $x=1$ .

From $x=1$ to $x=3$ , the slopes change from approximately $1$ to $-2$ . The slopes are decreasing, so the graph is concave down on

(1,4).

Step 3: Identify where concavity changes.

The graph changes from concave up to concave down at

x=1.

Therefore, $x=1$ is a possible inflection point.

The final description is:

\boxed{\text{Concave up on }(-4,1)}

\boxed{\text{Concave down on }(1,4)}

\boxed{\text{Possible inflection point at }x=1}

A local maximum or minimum does not automatically indicate a change in concavity. Always compare the slopes on the left and right sides of a point.

Learn by doing: Determine intervals of concavity from a graph

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Critical Points - Graph Concavity Feature to Interval

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