## Integer Power Functions

An example of an integer power function is:

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 **
**When x=1, y=1, since **
**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 **

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.

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 **
**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 **

We also have a few different rules

**If , then , 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 **
**For even functions as **
**For odd functions as **

**Note: ** means approaching zero from the positive side and from the negative side

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

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