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Convexity and Concavity

Convexity and Concavity

This article will explore convexity and concavity, both in functions and polygons. When we talk about convexity and concavity, we are referring to the shape of the curve or function. Now, convexity and concavity do sound like complicated terms. However, we will learn that they are nothing so fearful- they are simply a way of describing what a curve or function looks like. So, without any further introduction, let's define what exactly concave and convex functions are.

What is a Concave Function?

First, we will discuss concave functions.

A concave function is a function where a straight segment between any two points on the graph does not lie above the curve of the graph. In other words, the straight line is always below or on the curve.

Shown below is an example of a concave function. You can see that if we pick any two points on the curve, and draw a line segment between them, the line segment will always lie below the curve.

Convexity and Concavity, Example of a concave function, StudySmarter

Example of a concave function- StudySmarter Originals

Although this example only uses two points, for a function to be concave the rule must be true for all combinations of points on that function, in the given range.

How to find concavity in an equation

To algebraically portray that a function is concave, the following equation is used:

In other words, supposeandare any two points on the x-axis upon which the function is graphed.

Before breaking this down into words, it is important to understand thatselects any point between the pointsand. Similarly, selects any point betweenand.

The first portion of the inequality finds the value of the function of any point betweenand. The second portion selects any point between the functions of and . Thus, what this equation represents is that the function of any point betweenandis greater than or equal to any point between pointsand.

Observing the graph above, it is clear that this is true for that graph as the functions of points between the coordinates of intersection are above the functions between both functions, represented by the blue linear equation.

Remember that is any and all numbers between one and zero, such that all points and functions between the original parameters are verified.

What is a Convex Function?

A convex function is a function where a straight segment between any two points on the graph does not lie below the curve of the graph, In other words, the straight line is always above or at the same place as the function's curve. It is the opposite of a concave function.

An example can be seen below:

Convexity and Concavity, Example of a convex function, StudySmarter

Example of convex function- StudySmarter Originals

It is clear that this type of function opposes a concave function. A line between two points (representing all functions between both functions) is always above or at the same level as the function itself.

How to find Convexity in an Equation

Since both types of functions have similar parameters, their equations resemble each other. They have only one vital difference:

Notice the inequality sign is flipped the other way. Since all other components are identical, this function represents that any point between the selected coordinates is greater than or equal to any point on the function between both coordinates.

Can a Function be both Concave and Convex?

Yes, this is possible. This is because both functions have an equal sign in the inequality. The most common example of this is any straight line, as the function for a point between any two points will match the equivalent function between both functions.

What are Concave Polygons?

A concave polygon is any geometric shape where at least one internal angle exceeds 180 degrees (or radians). That is, there is a line that bends further inside than a straight line.

An example of this type of shape can be seen below:

Convexity and Concavity, example of concave polygon, StudySmarter

Example of concave polygon- StudySmarter Originals

In the above shape, the angle EDC exceeds 180 degrees. Therefore, it is a concave function.

There is a visual test that can be done to check for a concave polygon: if a straight line between any two points inside a polygon goes outside of the shape, that shape is a concave polygon. For example, below we can see that if we draw in the straight line segment , the line goes outside of the shape. Therefore, the polygon is concave.

Convexity and Concavity, Visual test for concave polygon, StudySmarter

Visual test for concave polygon- StudySmarter Originals

What are Convex Polygons?

A convex polygon is any polygon where no internal angle exceeds 180 degrees ( radians), that is, there is no internal angle that bends further than a straight line. Recall that a polygon is a shape made entirely of line segments.

An example would be the following shape:

Convexity and Concavity, convex polygon example, StudySmarter

Convex polygon example- StudySmarter Originals

The visual test for concave polygons can be inverted to test for a convex polygon. Since this polygon has no two points which create a segment that crosses outside of it, this geometric shape is a convex polygon.

Differences between concavity and convexity

The main difference between concavity and convexity is the fact that the angles subtended in convex shapes curve outwards whereas the angles subtended in concave shapes curve inwards. This is all based on whether or not there is an angle that exceeds 180 degrees.

Below are some further examples of concave and convex polygons. See if you can determine whether they are concave or convex.

For the following polygons, determine whether they are concave or convex.

Solution:

Convexity and Concavity, Determining concavity or convexity example 1, StudySmarter

Determining concavity or convexity example 1- StudySmarter Originals

In the above shape, we can see that there are interior angles that exceed 180 degrees. For example, the angle KJI exceeds 180 degrees. Thus, it is concave.

Convexity and Concavity, Determining concavity or convexity example 2, StudySmarter

Determining concavity or convexity example 2- StudySmarter Originals

For this polygon, we can see that there are also interior angles that exceed 180 degrees. For example, the angle EDC exceeds 180 degrees. Thus, it is concave.

Convexity and Concavity, Determining concavity or convexity example 3, StudySmarter

Determining concavity or convexity example 3- StudySmarter Originals

For the above polygon, we can see that there are no interior angles that exceed 180 degrees. Thus, it is convex.

Below are some further examples of concave and convex functions. See if you can determine whether they are concave or convex.

For the following functions, determine whether they are concave, convex or both.

Solution:

Convexity and Concavity, Determining concavity or convexity of functions example 1, StudySmarter

Determining concavity or convexity of functions example 1- StudySmarter Originals

For the above example, we have a cubic function. If we were to draw in the line segment from the point to the point , it would lie above the curve. Thus, this function is convex.

Convexity and Concavity, Determining concavity or convexity of functions example 2, StudySmarter

Determining concavity or convexity of functions example 2- StudySmarter Originals

Now, above we have a quartic function. We can see that any line segment drawn will lie below the curve. Thus, the function is concave.

Convexity and Concavity, Determining concavity or convexity of functions example 3, StudySmarter

Determining concavity or convexity of functions example 3- StudySmarter Originals

Finally, we have a straight line. Any line segment will lie on the line and hence it is both concave and convex.

Convexity and Concavity - Key takeaways

  • No segment created by any two points on a Concave Function will be above the function itself.
  • No segment created by any two points on a Convex Function will be below the function itself.
  • A function can be both Concave and Convex (e.g. a straight line).
  • A Concave polygon has an internal angle greater than 180 degrees.
  • A Convex polygon has no internal angle greater than 180 degrees.

Frequently Asked Questions about Convexity and Concavity

Yes. An example would be a straight line.

Convex: It is when a function is above or level with a straight segment between two points that lie on the function.

Concave: It is when a function is below or level with a straight segment between two points that lie on the function.

A concave polygon has at least one angle that exceeds 180 degrees. A convex polygon has no angles that exceed 180 degrees

 

If any line segment lies below the function, it is concave. If all line segments lie above or on the function, it is convex. 

In geometry, a concave polygon has at least one angle that exceeds 180 degrees. A convex polygon has no angles that exceed 180 degrees.

Final Convexity and Concavity Quiz

Question

Can a function be both concave and convex?

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Answer

Yes- a straight line is both concave and convex.

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Question

Which of the following would be a convex function?

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Answer

Both

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Question

Which of the following is a concave function?

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Answer

Neither

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Question

Which equation is both concave and convex?

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Answer

y=2x+7

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Question

A polygon with an internal angle greater than 180 degrees is:

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Answer

Concave

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Question

A triangle will always be a:

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Answer

convex polygon

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Question

If the graph of a function is always below a straight line joining two coordinates on the function, that function is:

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Answer

Convex

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Question

If the graph of a function is always above a straight line joining two coordinates on the function, that function is:

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Answer

concave

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Question

Which type of function is both concave and convex?

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Answer

Straight line functions 

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Question

Would a logarithmic graph be:

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Answer

Concave

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Question

Would a regular hexagon be concave or convex?

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Answer

Each interior angle of a regular hexagon is 120 degrees so it is concave. 

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Question

Could a quadrilateral be concave? 

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Answer

Yes- we could have one angle that is 190 degrees, one angle 100 degrees, one angle 10 degrees and one angle 60 degrees. 

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