Skill: 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.