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Concave: Definition, Shape & Function

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In mathematics, we can have concave shapes and concave functions. In this lesson, through definition and example, we will learn what it means to be concave and what these shapes and functions look like.

Definition of Concave

The definition of concave is to be curved inward. For instance, consider the following image of a fractured skull.

Concave Skull
concave 1

At the location of the fracture, we see the skull is curved inward, so we would say the skull is concave. Some other examples would be a dented can or car bumper. Whenever an object is curved inward, we say the object is concave.

In mathematics, the term concave can be applied to shapes and functions. Let's go over what it means for a shape to be concave and what it means for a function to be concave.

Concave Shape

We just saw the definition of concave is to be curved inward. This is also how we would describe a concave shape. A shape that is curved inward is a concave shape. The formal definition of a concave shape is a shape in which it's possible to draw two points within the shape and the line connecting the two points goes outside of the shape. This is illustrated in the image below:

Concave Shape
concave 2

Observe that the shape in the image above looks as though it's caved in. Because the name concave includes the word 'cave' in it, it's easy to remember that a concave shape is a shape that looks as though it's caved in somewhere.

When dealing with polygons (shapes created with three or more line segments), another rule exists that helps us to recognize a concave polygon. That is, a polygon is concave when at least one of its inside angles is greater than 180 degrees.

Let's consider a non-example, such as a circle, which is curved outward everywhere and does not look caved in anywhere. Look at the following image of a circle:

A Circle Is Not Concave
concave 3

Notice that no matter where you place two points within a circle, the line connecting the two points never goes outside the circle. Therefore, a circle is not concave; when a shape is not concave, we call it convex.

Concave Function

The concavity of a function has to do with the slope of a function. Recall that the slope of a function is the rate at which the function is increasing or decreasing and can be found by calculating the change in y divided by the change in x. When the slope of a function is increasing, we say the function is concave up, and when the slope of a function is decreasing, we say the function is concave down. This is illustrated in the following image:

Concave Up and Concave Down
Concave 4

Saying a function is concave is synonymous with saying a function is concave down. Therefore, a function is concave when the slope of the function is decreasing. For example, the function y = -x^2 is concave. In the following image, notice that the slope of y = -x^2 is always decreasing.

Graph of y = -x^2
concave 9

Examples

Consider the following shapes. Which of them are concave?

Example
concave 5

1.) In this image, we are able to draw two points within the shape and connect them with a line that goes outside the shape, as illustrated below. Therefore, the shape is concave.

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