Integer Power Functions

An example of an integer power function is:

$f(x)=x^{n}$

Where x can be any variable and n is a whole number either positive or negative

Power Function with a Positive Integer

This is where n would be equal to any positive whole number

Some examples of this are:

The basic shape of x to the power of an even number is a U shape, this we can see in the blue and black graphs above.

The basic shape of x to the power of an odd number is close to an S shape, this we can see in the red graph above.
For the non-transformed positive integer power functions we have a few rules:

When x=0, y=0, since $y=0^{n}=0$
When x=1, y=1, since $y=1^{n}=1$
When n is an even number, x=-1, y=1, since there are an equal amount of negatives, it makes a positive number
When n is an odd number, x=-1, y=-1, since there are an odd amount of negatives, it makes a negative number
As the value of n increases the graphs become steeper at higher values of x and become flatter when $0<x<1$

When multiple power functions are added together they become polynomials and we will describe sketching them in the quadratic, polynomial and cubic sections.

Power functions with a negative number

The second type of a power function is an inverse function where we have x to the power of any negative integer or one over x to the power of any positive integer.

$f(x)=x^{-n} or \frac{1}{x^{n}}$

Where n would be equal to any positive whole number

Let’s look at an even and odd example of this:

For the non-transformed negative integer power functions we have a few rules, which are the same as the positive integer power function:

When x=1, y=1, since $y=1^{n}=1$
When n is an even number, x=-1, y=1, since there are an equal amount of negatives, it makes a positive number
When n is an odd number, x=-1, y=-1, since there are an odd amount of negatives, it makes a negative number

Then we have one rule which is the opposite of the positive integer power function:

As the value of n increases the graphs become flatter at higher values of x and become steeper when $0<x<1$

We also have a few different rules

If $f(x)=\frac{1}{x^{n}}$ , then $x \neq 0$ , therefore the domain for any negative power integer functions is R\{0} and we have an assymptote at x=0
For both even and odd functions as $x\rightarrow0^{+},y\rightarrow\infty^{+}$
For even functions as $x\rightarrow0^{-},y\rightarrow\infty^{+}$
For odd functions as $x\rightarrow0^{-},y\rightarrow\infty^{-}$

Note: $0^{+}$ means approaching zero from the positive side and $0^{-}$ from the negative side

Once these basic rules have been mastered transformations can be applied later on to graph more complex functions.

Feedback

Want to suggest an edit? Have some questions? General comments? Let us know how we can make this resource more useful to you.

While we endeavour to provide you with great study material - we’re not qualified teachers, as such Engage cannot guarantee the validity of the information here.

The Victorian Curriculum and Assessment Authority (VCAA) does not endorse this website and makes no warranties regarding the correctness or accuracy of its content. VCE is a registered trademark of the VCAA. Current and past VCE exams and related content can be accessed directly at www.vcaa.vic.edu.au

Contents