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How to Analyze Power Functions

Power functions are a special case of polynomial functions. Power functions only have one term—a polynomial in the form axn.

Power functions are analyzed in the same way as normal polynomial functions. Here is the method:

Rule

Analyzing Power Functions

1.
Find the zeros.
2.
Find the stationary points.
3.
Find the inflection points.

Example 1

Analyze the function f(x) = 5x4

1.
First, you find the zeros by setting f(x) = 0. You then get: 5x4 = 0|÷ 5 x4 = 0 x44 = 04 x = 0

So you have a zero at (0, 0).

2.
You then find the maxima and minima by setting f(x) = 0. First, you differentiate the function:
f(x) = 20x3

You then set this equal to 0 to find the maxima and minima: 20x3 = 0 x = 0

To determine if it is a maximum or a minimum, you can select two values, one to the left of x = 0 and one to the right of x = 0. Input these into the differentiated function f(x) and interpret the sign. Choose numbers that are easy to work with, like x = 1 and x = 1:

f(1) = 20(1)3 = 20 < 0 graph decreases f(1) = 20(1)3 = 20 < 0 graph increases

f(1) = 20(1)3 = 20 < 0 graph decreases f(1) = 20(1)3 = 20 < 0 graph increases

As the graph decreases first and then increases, this is a minimum.

You now need to find the y-value of the point, by inserting the x-value into the main function f(x):

f(0) = 5(0)4 = 0

Thus, you have a minimum at (0, 0).

3.
To find the inflection points of the function, you set f(x) = 0. First, you find f(x):
f(x) = 60x2

Set this equal to 0 and you get: 60x2 = 0 x = 0

You can now find the y-value of the inflection point by inserting your x-value into the main function f(x):

f(0) = 5(0)4 = 0

As you can see from this, the zero, the inflection point and the minimum are all the same point.

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