Fact: The graph of a rational function p(x)/q(x) (in reduced terms) will be have no jumps except at the zeros of q(x), where it shoots off to ±.

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1 Rational functions Some of these are not polynomials. 5 1/x 4x 5 + 4x 2 x+1 x 1 (x + 3)(x + 2)() Nonetheless these non-polynomial functions are built out of polynomials. Maybe we can understand them in terms of our understanding of polynomials. Definition 1. A rational function is a ratio of polynomials. That is, anything of the form p(x) q(x) with p(x) and q(x) polynomial is called a rational function. Which are rational functions? x2 x x x 3 1. x + x 1. x + 5 x 3 1 x x + 1 (x3 1)(x + 1). x 3 + x + 1. x. Just like we would simplify 2 4 to 1 2 we will always cancel common factors. So, (x 2 + 1)(x 2) (x 2)(x + 1) = graphs of rational functions Here is the graph of 1/x How does this differ from the graph of a polynomial? What is the domain of 1/x? Fact: The graph of a rational function p(x)/q(x) (in reduced terms) will be have no jumps except at the zeros of q(x), where it shoots off to ±. 1

2 2 An asymptote for a curve is a line for which the distance between the line and curve gets small as we look further and further along the line. Here is the graph of 1/x. Find its asymptotes. 1/x has a vertical asymptote at. It has a horizontal asymptote at. Understanding check Sketch a graph which has a horizontal asymptote at y = 2 and vertical asymptotes at x = 3 and x = 2. Results may vary. Facts: If p(x)/q(x) is in reduced form, then p(x)/q(x) has an x-intercept whenever If p(x)/q(x) is in reduced form, then p(x)/q(x) has a vertical asymptote whenever Where are the x-intercepts and vertical asymptotes of the following rational functions? x + 1 x2 1 x x2 + 3

3 3 Long division and graphing Perform polynomial long division on x + 1 x + 1 = to get How can we get the graph of this starting from the graph of 1 x? Double Check: Does this have the x-intercepts and vertical asymptotes which we know it must? Exercise: Using long division, put x + 3 into the form x 2 2x + 3 x 2 = A + B x 2 Graph this starting from the graph of 1 x. A Check sum: Are the x-intercepts and vertical asymptotes where they should be?

4 4 The horizontal asymptotes of a rational function can be read off: Theorem 2. For the rational function p(x)/q(x). If the degree of q is larger than the degree of p then there is a horizontal asymptote at y =. If the degree of q is equal to the degree of p then there is a horizontal asymptote at ( ) y = ( ). If the degree of p is greater than the degree of q then Sanity check Does this theorem agree with the graphs we ve generated? Example: Where are the horizontal asymptotes if there are any? x 4 + x x x 3 x5 1 x x4 + x x 5 x x Example: The bee population (in hundreds of nests) in a randomly chosen wood is modeled by random population = 5t t In the hundred acre woods it is modeled instead by 100 acre population = 4t t As time goes on to infinity, where do these bee populations settle?

5 5 We will give two different strategies for graphing rational functions: (x + 1)(x 2) Graphing: Let s sketch a graph of (x + 2)() which has the correct x and y-intercepts (Find these) has the correct vertical asymptotes (find these) has the correct horizontal asymptote (Find it) Is positive and negative in the correct places 2()(x + 3) Graphing: Let s sketch a graph of (x + 2)(x 2) which has the correct x and y-intercepts (Find these) has the correct vertical asymptotes (find these) has the correct horizontal asymptote (Find it) has the correct behavior near the x-intercepts and vertical asymptotes.