Time to put all of it together. Here are some steps that sum up the video for graphing rational functions:

1) Find any intercepts. Remember to find x-intercepts by setting the factors in the numerator equal to zero. To find any y-intercepts evaluate the whole function for \(x=0\).

2) Find the vertical asymptotes by setting the denominator equal to zero and solving.

3) Find any horizontal asymptotes.

4) Draw the asymptotes using dotted lines. The vertical asymptotes divide the graph into sections.

5) Find some points in each section and sketch the graph.

Pick two of the functions below.

1) Find all the key features we have discussed (asymptotes, holes, intercepts).

2) Sketch a graph of the function.

3) Check answers with Desmos or another graphing utility. Key features are on solution page.

**desmos.com/calculator**

a) \(f(x)=\Large \frac{1}{3x^2+3x-18}\)

b) \(f(x)=\Large \frac{x-3}{x-4}\)

c) \(f(x)=\Large \frac{x^3-x^2-8}{-4x^2-4x+24}\)

d) \(f(x)=\Large \frac{x-5}{-5x-20}\)

e) \(f(x)=\Large \frac{4x^2-16x}{x^2-3x-4}\)

f) \(f(x)=\Large \frac{3x^2+15x+18}{x^2+5x+6}\)

Check solutions here.

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