What is a Power Function? - Definition, Equations, Graphs & Examples

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  • 0:01 Power Functions
  • 2:36 Graphs of Power Functions
  • 5:17 Fractional Power Functions
  • 7:45 Lesson Summary
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Lesson Transcript
Instructor: Cathryn Jackson

Cat has taught a variety of subjects, including communications, mathematics, and technology. Cat has a master's degree in education and is currently working on her Ph.D.

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 nth power of x. This basically means that the two graphs would look the same. Here is a graph showing x^4:

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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.

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If n is less than zero, then the function is inversely proportional to the nth 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.

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In a power function, k represents the constant of proportionality. This means that the shape of the line on the graph will not change depending on the value of k, but the placement of the line on the graph will change. Take a look at this graph to see what I mean.

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The blue line on this graph is the equation f(x) = x^3, and the green line is the equation f(x) = 5x^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.

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This is the graph of f(x) = -x^2. Here k = -1 and n = 2. Notice that this graph is the opposite of the first graph you saw. The only thing that changed in the equation is the negative sign on the k value. Often, the negative sign will indicate the opposite or reverse.

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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.

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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.

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Savanna is studying comets that go straight towards the earth, veer off to the side, and then keep going in a straight line past the earth. She is probably looking at comets that make the path f(x) = x^3. Take a look at this graph and see if it matches Savanna's description.

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

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Wow! Notice that, like the other graphs that had negative exponents, the lines on the graph sort of separate into two different directions.

However, unlike the graph f(x) = x^-4, shown with blue lines, the graph of f(x) = x^-3, shown with green lines, has negative y-coordinates. This is because the power in this function is odd, which will give you a negative result.

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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.

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