# Functions, Graphs, and Limits

### Topics

## Introduction to Functions, Graphs, And Limits - At A Glance:

We now have enough tools to draw some complicated functions from scratch. Now we know how graphing calculators do it, and why they require the energy of four triple-A's.

When drawing a rational function *f*(*x*) from scratch, we need to know a lot of information, which can be nicely grouped into three big chunks.

- We need to know where
*f*has vertical asymptotes and/or holes. - We need to know the horizontal/slant/curvilinear asymptotes of
*f*, if any. - We need to know about values of
*f*. We found where*f*is undefined when we found the vertical asymptotes and holes; now we need to know where*f*(*x*) is 0, positive, and negative. We also want to know*f*(0), also called the*y*-intercept.

#### Example 1

Graph the function |

#### Example 2

Graph the function |

#### Example 3

Graph the function |

#### Example 4

Graph the function Label all asymptotes, intercepts, and holes. |

#### Exercise 1

Graph the function. Label all asymptotes, intercepts, and holes.

#### Exercise 2

Graph the function. Label all asymptotes, intercepts, and holes.

#### Exercise 3

Graph the function. Label all asymptotes, intercepts, and holes.

#### Exercise 4

Graph the function. Label all asymptotes, intercepts, and holes.

#### Exercise 5

Graph the function. Label all asymptotes, intercepts, and holes.