Power functions have many different varieties of graphs. Use this lesson to distinguish the difference between different types of power functions. Test your knowledge with a short quiz after the lesson!

## Power Functions

Savanna is studying the paths of asteroids, comets, and other bodies that fly through space. She notices that as a certain comet gets closer to the earth, the path of the comet curves and moves away.

Savanna wants to create mathematical functions of the comet and asteroid paths as they come closer to the earth. This way, she can predict the paths of certain comets and asteroids. Savanna can use her knowledge of power functions to create equations based on the paths of the comets.A **power function** is in the form of *f(x)* = *kx*^*n*, where *k* = all real numbers and *n* = all real numbers. You can change the way the graph of a power function looks by changing the values of *k* and *n*.If *n* is greater than zero, then the function is proportional to the *n*th power of *x*. This basically means that the two graphs would look the same.

Here is a graph showing *x*^4:

So in this graph, *n* is greater than zero. Here is the graph of *f(x)* = *x*^4.

There is no difference between the two graphs.

If *n* is less than zero, then the function is inversely proportional to the *n*th power of *x*. That means you will see the graph sort of flipped. Let’s look at our graph of *x*^4 again.

Now let’s look at the graph of *x*^-4. Notice that this graph has an empty space near the origin. It almost creates a cut-out section.

The blue line on this graph is the equation *f(x)* = *x*^3, and the green line is the equation *f(x)* = 5*x*^3. Notice that when we add the 5 in front of the *x*, the shape of the graph stays the same, but the line moves closer to the origin.

These concepts will become more important as you study calculus, but you do need to keep them in mind as you explore power functions.

## Graphs of Power Functions

This is the graph of *f(x)* = *x*^2. You’ve probably seen this type of function a lot; the shape it creates is a parabola. In this graph, *k* = 1 and *n* = 2.

These types of functions, functions that contain the *x*^2 value, are called quadratic functions.Here is a graph of the function *f(x)* = *x*^4. Notice that the bottom of this function sort of widens, but it never crosses over the *x*-axis.

This means that all of the *y*-coordinates of this function are positive.

Let’s look what happens when we make *n* negative in this function. This time the lines on the graph split into two sections, but the line still does not cross the *x*-axis. This is the graph of *f(x)* = *x*^-4.

Looks pretty close, huh? What about the function *f(x)* = *x*^-3? What would that graph look like?

You’ll notice that functions with an even power are symmetrical across the *y*-axis and functions with an odd power are symmetrical about the origin.

You can learn more about symmetry in the Graph Symmetry chapter of this course.

## Fractional Power Functions

Savanna is now studying the path of asteroids. This time, she needs to find the difference between the asteroids that go past the earth and the asteroids that collide with the earth. She has two different asteroids she is studying. The first has the function of *f(x)* = *x*^1/3, and the second asteroid has the function of *f(x)* = *x*^1/4.

This is the graph of *f(x)* = *x*^1/3. You’ll see that the line on this graph goes from the positive side, curves, and then to the negative side of the graph. We can assume that the asteroid that creates this path will not collide with the earth.

This is the graph of *f(x)* = *x*^1/4. You’ll see that the line on this graph goes from the positive side and stops at the origin. We can assume that the asteroid that creates this path will collide with the earth.

Now that we’ve looked at positive fractional power functions, let’s look at some negative fractional power functions.The first graph, represented with green lines, is the function *f(x)* = *x*^-3/5. The second graph, represented with a blue line, is the function *f(x)* = *x*^-1/4. Notice that the function with the even denominator is located only on the positive side of the *x*– and *y*-axis.

Notice that these two functions, *f(x)* = *x*^-4 and *f(x)* = *x*^-1/4, look very similar.

Notice that the only differences in these graphs are the positions of the curves of the lines. You’ll see that all of the numbers in the powers of the two functions are odd numbers.

Lastly, we must consider the functions that have powers with improper fractions, such as this graph. Notice that this graph does not contain any negative *x*– or *y*-coordinates. This graph represents the function *f(x)* = *x*^5/2.

Now that you’ve seen all of these graphs, how can you remember which functions go with which? Let’s summarize.

## Lesson Summary

A **power function** appears in the form *f(x)* = *kx*^*n*, where *k* = all real numbers and *n* = all real numbers. The three main types of power functions are even, odd, and fractional functions.

You will see these in both positive and negative forms.Here are the graphs of the functions *f(x)* = *x*^2 and *f(x)* = *x*^4. These graphs are shaped similarly because they both have positive and even powers.

These graphs are similar because they both contain positive, odd numbers in their powers.

The first graph represents the function *f(x)* = *x*^3, and the second graph represents *f(x)* = *x*^5/2.

Notice these graphs have a nice sloping curve close to the origin. These graphs are similar because they all have positive, fractional powers.

Also, don’t forget when you are graphing power functions to graph with a curvy line.

## Learning Outcomes

Watch this video lesson as you pursue these goals:

- Define and use power functions
- Recall the equation form for a power function and the write down three main types
- Accurately identify whether a graph is an even or odd powered function
- Graph a power function