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Inflection Point: Simple Definition & Examples

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You may want to read about concave up and concave down first.


What is an Inflection Point?

An inflection point (sometimes called a flex or inflection) is where a graph changes curvature, from concave up to concave down or vice versa.

A concave up graph is like the letter U (or, a “cup”), while a concave down graph is shaped like an upside down U, or a Cap (∩). The inflection point happens when a “cup” and a “cap” meet.
inflection point

Vertical Inflection Point

vertical inflection points

A vertical inflection point, like the one in the above image, has a vertical tangent line; It therefore has an undefined slope and a non-existent derivative.
At first glance, it might not look like there’s a vertical tangent line at the point where the two concavities meet. However, if you zoom in close enough (perhaps with a graphing calculator), you’ll see that there’s a tiny, almost insignificant spot in the graph where the tangent line is perfectly vertical.
vertical tangent line

Horizontal Points of Inflection

A horizontal point of inflection is where the tangent line is horizontal. The derivative or slope here is zero, because the tangent line is flat.

horizontal inflection point

The horizontal inflection point (orange circle) has a horizontal tangent line (orange dashed line).

Inflection Points and Rates of Change

You can also think of an inflection point as being where the rate of change of the slope changes from increasing to decreasing, or increasing to decreasing.

When slopes of tangent lines increase (from left to right, regardless of sign), your graph is concave up:
slope of tangent line

On the other hand, concave down graphs have decreasing slopes, from left to right:

So if you know what the rates of change are at any point on your graph, you can tell where there’s an inflection point.

Using the Second Derivatives

The second derivative test uses that information to make assumptions about inflection points. The next graph shows x3 – 3x2 + (x – 2) (red) and the graph of the second derivative of the graph, f” = 6(x – 1) in green. Positive x-values are to the right of the inflection point; Negative x-values are to the left.
second derivative test concavity

Stephanie Glen. "Inflection Point: Simple Definition & Examples" From Calculus for the rest of us!

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One thought on “Inflection Point: Simple Definition & Examples

  1. norman mason

    My favourite example is the inflection point at the origin on the curve y = cube root of x.

    so simple to see and it doesn’t require d2y/dx2 to be zero or dy/dx to be defined at the point

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