Concavity (College Board AP® Precalculus): Study Guide

Concavity of a function

How is rate of change defined for a non-linear graph?

• On the graph of a non-linear function (i.e. a curve rather than a straight line)

  • the rate of change at a point on the graph

  • is defined as the slope of the straight tangent line drawn at that point

• If the slope of the tangent line is positive then the rate of change is positive

  • This occurs when the function is increasing

• If the slope of the tangent line is negative then the rate of change is negative

  • This occurs when the function is decreasing

• The rate of change is greater when the graph is steeper

• E.g. in the image below

  • tangents drawn at points A and B show that the rate of change is positive at both those points

  • But the graph is steeper at B

  • therefore the rate of change at B is greater

What does it mean for the graph of a function to be concave up?

• The graph of a function is said to be concave up on an interval where the rate of change is increasing

  • This may mean that a positive rate of change is becoming more positive

  • or that a negative rate of change is becoming less negative

• A concave up section on a graph is often 'cup' or -shaped

What does it mean for the graph of a function to be concave down?

• The graph of a function is said to be concave down on an interval where the rate of change is decreasing

  • This may mean that a positive rate of change is becoming less positive

  • or that a negative rate of change is becoming more negative

• A concave down section on a graph is often 'upside down cup' or -shaped

What is a point of inflection?

• A point of inflection is a point where the graph of a function

  • changes from concave down to concave up

  • or changes from concave up to concave down