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.
Millions of flashcards designed to help you ace your studies
Sign up for freeWhich of the following would be a convex function?
Which of the following is a concave function?
Would a logarithmic graph be:
Which equation is both concave and convex?
A polygon with an internal angle greater than 180 degrees is:
Would a regular hexagon be concave or convex?
A triangle will always be a:
If the graph of a function is always below a straight line joining two coordinates on the function, that function is:
If the graph of a function is always above a straight line joining two coordinates on the function, that function is:
Which of the following would be a convex function?
Which of the following is a concave function?
Would a logarithmic graph be:
Which equation is both concave and convex?
A polygon with an internal angle greater than 180 degrees is:
Would a regular hexagon be concave or convex?
A triangle will always be a:
If the graph of a function is always below a straight line joining two coordinates on the function, that function is:
If the graph of a function is always above a straight line joining two coordinates on the function, that function is:
Achieve better grades quicker with Premium
Geld-zurück-Garantie, wenn du durch die Prüfung fällst
Review generated flashcards
Start learning or create your own AI flashcards
Team Convexity and Concavity Teachers
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.
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.
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.
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:
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.
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.
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.
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:
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.
Visual test for concave polygon- StudySmarter Originals
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:
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.
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:
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.
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.
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:
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.
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.
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.
Sign up for free to gain access to all our flashcards.
Can a function be both concave and convex?
Yes. An example would be a straight line.
What is concave and convex in maths?
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.
What is concave and convex shapes?
A concave polygon has at least one angle that exceeds 180 degrees. A convex polygon has no angles that exceed 180 degrees
How to find concavity and convexity of a function?
If any line segment lies below the function, it is concave. If all line segments lie above or on the function, it is convex.
What does concave and convex mean in geometry?
In geometry, a concave polygon has at least one angle that exceeds 180 degrees. A convex polygon has no angles that exceed 180 degrees.
At StudySmarter, we have created a learning platform that serves millions of students. Meet the people who work hard to deliver fact based content as well as making sure it is verified.
Lily Hulatt is a Digital Content Specialist with over three years of experience in content strategy and curriculum design. She gained her PhD in English Literature from Durham University in 2022, taught in Durham University’s English Studies Department, and has contributed to a number of publications. Lily specialises in English Literature, English Language, History, and Philosophy.
Get to know Lily
Gabriel Freitas is an AI Engineer with a solid experience in software development, machine learning algorithms, and generative AI, including large language models’ (LLMs) applications. Graduated in Electrical Engineering at the University of São Paulo, he is currently pursuing an MSc in Computer Engineering at the University of Campinas, specializing in machine learning topics. Gabriel has a strong background in software engineering and has worked on projects involving computer vision, embedded AI, and LLM applications.
Get to know Gabriel