2 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:
3 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!!!!
4 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!!!
5 Note: the graph represents a hyperbola centered at (0, 0) xyY−33−22−11−.5.5-.3333.3333-.5.5-11-22In 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)
6 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)!!
7 Vertical asymptote: x = –3 Horizontal asymptote: y = 2Show your asymptotes!!Pick some x’s on each side of the vertical asymptote to see the graph!!!xyY−4–2−5–1413Use more points if you want . . .Df = (, 3), (3, )Rf = (, 2), (2, )
8 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 = –2y = 3xyY−3–1−48–25.5.5Df = (, 2), (2, )Rf = (, 3), (3, )
10 x = 4; y = 1Remember: 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!!!