# Section 3.2 - Basic Rational & Power Functions and More Transformations

**Section Objectives**

- Develop a familiarity with the whole-number power functions and basic rational functions.
- Apply transformations to basic functions.

### Intro to Polynomials and Rational Functions

An *-th-degree polynomial* in the variable is a function of the form

,

where the coefficients , , , are real or complex numbers with . We will be almost exclusively interested in polynomials with real coefficients.

Here are some examples of polynomial functions...

A *rational function* is a ratio of two polynomials. That is, a rational function is a function of the form

where and are polynomials and .

The domain of a rational function is the set of all real numbers for which the denominator is nonzero.

Here are some examples of rational functions...

### Reciprocal functions

One of the simplest rational functions is the reciprocal function .

- Domain
- Range
- The reciprocal function is an odd function. Its graph is symmetric about the origin.
- As , . (The -axis is a horizontal asymptote of the graph.)
- As , . (The -axis is a vertical asymptote of the graph.)

Another of the simpler rational functions is the reciprocal square function .

- Domain
- Range
- The reciprocal square function is an even function. Its graph is symmetric about the -axis.
- As , . (The -axis is a horizontal asymptote of the graph.)
- As , . (The -axis is a vertical asymptote of the graph.)

#### Examples

- Sketch the graph of . Domain? Range? Asymptotic behavior?

- Determine the horizontal and vertical asymptotes of the graph of .

### Even-number Power Functions: ,

The graphs of , , , ... are U-shaped curves opening upward with the turning point at the origin.

The bigger the exponent, the flatter the curve is on the interval and the steeper the curve is outside that interval.

### Odd-number Power Functions: ,

The graphs of , , , ... have the following shape. Each has a flat spot at the origin.

The bigger the exponent, the flatter the curve is on the interval and the steeper the curve is outside that interval.

#### Examples

- Sketch the graph of . Domain? Range? End behavior?

- Sketch the graph of . Domain? Range? End behavior?

- The graph of is shifted downward by 7 units. Find an equation for the new graph.

- Let .

(1) Describe the graph of .

(2) Find the graph's intercepts.

(3) Find the coordinates of the vertex.

(4) Find an equation for the symmetry axis.

- Describe and sketch the graph of .