1. Graphs of Rational Functions Section 2.7

2. Objectives Analyze and sketch graphs of rational functions. Sketch graphs of rational functions that have slant asymptotes.

3. Let define a rational expression in lowest terms. To sketch its graph, follow these steps. 1.Find all asymptotes. 2.Find the x- and y-intercepts. 3.Determine whether the graph will intersect its non- vertical asymptote by solving f (x) = k where y = k is the horizontal asymptote, or f (x) = mx + b where y = mx + b is the equation of the oblique asymptote. 4.Plot a few selected points, as necessary. Choose an x- value between the vertical asymptotes and x-intercepts. 5.Complete the sketch. Graphing Rational Functions

4. Graphing a Rational Function with a Hole Example Graph SolutionNotice the domain of the function cannot include 2. Rewrite f in lowest terms by factoring the numerator. The graph of f is the graph of the line y = x + 2 with the exception of the point with x-value 2.

5. Graph Solution Step 1 Step 2x-intercept: solve f (x) = 0 Example 1

6. y-intercept: evaluate f (0) Step 3To determine if the graph intersects the horizontal asymptote, solve Since the horizontal asymptote is the x-axis, the graph intersects it at the point (–1,0). Example 1

7. Step 4Plot a point in each of the intervals determined by the x-intercepts and vertical asymptotes, to get an idea of how the graph behaves in each region. Step 5Complete the sketch. The graph approaches its asymptotes as the points become farther away from the origin. Example 1

8. ExampleGraph SolutionVertical Asymptote: Horizontal Asymptote: x-intercept: y-intercept: Does the graph intersect the horizontal asymptote? Example 2

9. To complete the graph of choose points (–4,1) and. Example 2

10. Your Turn: Sketch the graph of y-intercept (0,0) x-intercept (0, 0) Vertical asymptotes; x=-1, x=2 Horizontal asymptotes; y=0 Additional points; (-3, -.3) (-.5,.4) (1, -.5) (3,.75)

11. Your Turn: Sketch the graph of y-intercept (0, 3) x-intercept (-3, 0) Hole (3, 3/2) Vertical asymptote x=-1 Horizontal asymptote y=1 Additional points (-5,.5) (-2, -1) (-.5, 5) (1, 2)

12. Slant Asymptotes Some asymptotes that are neither vertical or horizontal => they are slanted. These slanted asymptotes are called oblique asymptotes. Ex. Graph the function f(x) = (x 2 - x - 6)/(x - 2) (which brings us back to our previous work on the Factor Theorem and polynomial division) Recall, that we can do the division and rewrite f(x) = (x 2 - x - 6)/(x - 2) as f(x) = x + 1 - 4/(x - 2). Again, all we have done is a simple algebraic manipulation to present the original equation in another form.

13. Slant Asymptotes So now, as x becomes infinitely large (positive or negative), the term 4/(x - 2) becomes negligible i.e. = 0. So we are left with the expression y = x + 1 as the equation of the slant asymptote. Slant asymptotes occur when the degree of the denominator is 1 or greater and the degree of the numerator is exactly one more than the degree of the denominator. Find the equation of the slant asymptote by dividing the denominator into the numerator, using either synthetic or long division. The equation of the slant asymptote is then the quotient and disregard the remainder.

14. Graphing a Rational Function with an Oblique Asymptote ExampleGraph SolutionVertical asymptote: Oblique asymptote: x-intercept: None since x 2 + 1 has no real solutions. y-intercept:

15. Graphing a Rational Function with an Oblique Asymptote Does the graph intersect the oblique asymptote? To complete the graph, choose the points

16. Your Turn: Sketch the graph of y-intercept (0, 0) x-intercepts (0, 0) (1, 0) Vertical asymptote x=-1 Horizontal asymptote: none Slant asymptote y=x Additional points (-2, -4/3) (3, 2)

17. Assignment Pg 161 – 164: #9 – 25 odd, 43 – 49 odd, 61, 63 Review pg 175 – 178: #75 – 119