## 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

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

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.