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Functions, Graphs, and Limits

Functions, Graphs, and Limits

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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.

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