## Based on slope of graph. Increasing – concave up, decreasing – concave down

### Find where a function is concave down/up given a graph of f ‘(x).

The concavity of a graph is related to the second derivative of the function that defines the graph. When the second derivative is positive, the graph is said to be concave up, while when the second derivative is negative, the graph is said to be concave down.

The slope of a graph is related to the first derivative of the function that defines the graph. The slope of the graph is increasing when the first derivative is positive and decreasing when the first derivative is negative.

If the slope of the graph is increasing, it means that the tangent lines to the graph are getting steeper as we move along the x-axis. This usually indicates that the graph is curving upwards, which is consistent with concave up behavior.

On the other hand, if the slope of the graph is decreasing, it means that the tangent lines to the graph are getting flatter as we move along the x-axis. This usually indicates that the graph is curving downwards, which is consistent with concave down behavior.

Therefore, based on the slope of the graph, we can make an inference about the concavity of the graph. Increasing slopes are indicative of concave up behavior, while decreasing slopes are indicative of concave down behavior.

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