387 3.5 Rational Functions: Graphs, Applications, and Models (c) Setting the denominator x - 2 equal to 0 shows that the vertical asymptote has equation x = 2. If we divide by the greatest power of x as before (x2 in this case), we see that there is no horizontal asymptote because ƒ1x2 = x2 + 1 x - 2 = x2 x2 + 1 x2 x x2 - 2 x2 = 1 + 1 x2 1 x - 2 x2 does not approach any real number as x S∞, due to the fact that 1 + 0 0 - 0 = 1 0 is undefined. This happens whenever the degree of the numerator is greater than the degree of the denominator. In such cases, divide the denominator into the numerator to write the expression in another form. We use synthetic division, as shown in the margin. The result enables us to write the function as follows. ƒ1x2 = x + 2 + 5 x - 2 For very large values of x , 5 x - 2 is close to 0, and the graph approaches the line y = x + 2. This line is an oblique asymptote (slanted, neither vertical nor horizontal) for the graph of the function. This supports procedure 2(c) of determining asymptotes. S Now Try Exercises 37, 39, and 41. Graphing Techniques A comprehensive graph of a rational function will show the following characteristics. • all x- and y-intercepts • all asymptotes: vertical, horizontal, and/or oblique • the point at which the graph intersects its nonvertical asymptote (if there is any such point) • the behavior of the function on each domain interval determined by the vertical asymptotes and x-intercepts 2)1 0 1 2 4 1 2 5 Setup for synthetic division Graphing a Rational Function Let ƒ1x2 = p1x2 q1x2 define a function where p1x2 and q1x2 are polynomial functions and the rational expression is written in lowest terms. To sketch its graph, follow these steps. Step 1 Find any vertical asymptotes. Step 2 Find any horizontal or oblique asymptotes. Step 3 If q102 ≠0, plot the y-intercept by evaluating ƒ102. Step 4 Plot the x-intercepts, if any, by solving ƒ1x2 = 0. (These will correspond to the zeros of the numerator, p1x2.) Step 5 Determine whether the graph will intersect its nonvertical asymptote y = b or y = mx + b by solving ƒ1x2 = b or ƒ1x2 = mx + b. Step 6 Plot selected points, as necessary. Choose an x-value in each domain interval determined by the vertical asymptotes and x-intercepts. Step 7 Complete the sketch.
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