Rational Functions

Definition: A Rational Function is a function in the form: f(x) = where p(x) and q(x) are polynomial functions and q(x) ≠ 0. In this section, p and q will have degree 1 or 0. For example:

Very Important definitions: Vertical asymptote occurs at values of x for which the function is undefined (exception: unless there is a hole . . . we’ll talk about that later). Horizontal asymptote occurs if the function approaches a specific value when x approaches infinity or negative infinity. Think of the warmup: what happens to y when x gets REALLY big or REALLY small? To graph rational functions, ALWAYS figure out the asymptotes FIRST. Then you can plot specific points!!!!

Vertical asymptote will occur at x = 0. Consider: Vertical asymptote will occur at x = 0. Df = (, 0), (0, ) Horizontal asymptote will occur at y = 0. Think what happens when you divide 1 by a VERY large number!!!!! Show your asymptotes!! Pick some x’s on each side of the vertical asymptote to see the graph!!!

Note: the graph represents a hyperbola centered at (0, 0) x y Y −3   3 −2 2 −1 1 −.5 .5 -.3333 .3333 -.5 .5 -1 1 -2 2 In most cases, the range will be closely related to the horizontal asymptote . . . be sure to check the graph. Rf = (, 0), (0, ) Note: the graph represents a hyperbola centered at (0, 0)

Another type of rational function: The vertical asymptote is still x = h. Df = (, h), (h, ) Based on our observations, the hortizontal asymptote is y = k. So, this will be a hyperbola centered at (h, k)!!

Vertical asymptote: x = –3 Horizontal asymptote: y = 2 Show your asymptotes!! Pick some x’s on each side of the vertical asymptote to see the graph!!! x y Y −4   –2 −5 –1 4 1 3 Use more points if you want . . . Df = (, 3), (3, ) Rf = (, 2), (2, )

Last example: Asymptotes: x = –2 y = 3 8 –2 5.5 .5 Vertical asymptote will correspond to the value that makes the denominator 0. Horizontal asymptote: y = Asymptotes: x = –2 y = 3 x y Y −3   –1 −4 8 –2 5.5 .5 Df = (, 2), (2, ) Rf = (, 3), (3, )

x = 0; y = 0 x = 3; y = 1

x = 4; y = 1 Remember: Find the asymptotes FIRST. Show them on the graph!!! Pick x values to the right and to the left of the vertical asymptote(s). Use the points along with the asymptotes to sketch the graph!!!