What is an inflection point in a graph?

An inflection point in a graph is defined as a point where the concavity changes. This means that the graph transitions from being concave up (where it curves upwards) to concave down (where it curves downwards), or vice versa. At an inflection point, the second derivative of a function is typically equal to zero or does not exist, which indicates a change in the nature of the curve's steepness.

Understanding concavity is crucial in analyzing the behavior of functions, as it helps determine the shape of the graph and the nature of its turning points. This is particularly important for identifying key features in calculus and optimization problems, where recognizing inflection points can inform about the function’s growth and decay rates.

The other options represent different concepts related to graph behavior. Intersecting the x-axis refers to the roots of the function, maximum or minimum points denote local extrema, and points of continuity highlight where the function remains uninterrupted. None of these concepts address the change in concavity that characterizes inflection points.