1. Mrs.Volynskaya Ch.2.6 Rational Functions and Their Graphs

2. A rational function is a function that can be expressed in the form where both f(x) and g(x) are polynomial functions. Examples of rational functions would be:.

3. Looking at the graph, as the x values get larger and larger in the negative direction, the y values of the graph appear to get closer and closer to what? If you guessed that the y values appear to get closer and closer to 0, you may be onto something. Let’s look at a table of values for confirmation.

4. Vertical Asymptote at x = 2 Horizontal Asymptote at y = 0. It is very important to know where your asymptotes are before you start plotting points.

5. Next up is the graph of one of the functions that was mentioned back in frame #2. We have a vertical asymptote at x = 3 because at that value of x, the denominator is 0 but the numerator is not. Congratulations if you picked this out on your own.

6. Locating Vertical Asymptotes Ifis a rational function in which p(x) and q(x) have no common factors and a is a zero of the denominator, then x = a is a vertical asymptote of the graph of f.

7. Locating Horizontal Asymptotes Let f be the rational function given by The degree of the numerator is n. The degree of the denominator is m. 1.If n<m, the x-axis, or y=0, is the horizontal asymptote 2.If n=m, the line y = a n /b m is the horizontal asymptote 3.If n>m,the graph has no horizontal asymptote.

8. Here is the graph with most of the points in our table. Vertical asymptote at x = 3. Horizontal asymptote at y = -2.

9. Example 3 It appears that we may have vertical asymptotes at x = 0 and at x = 2. We will see if the table confirms this suspicion.

10. There are NO vertical asymptote at x = 2 because when x = 2 both the numerator and the denominator are equal to 0. Check this out: Does this mean thatand are identical functions? Yes, at every value of x except x = 2 where the function is undefined. There will be a hole in the graph where x = 2.

11. To summarize then, we have a vertical asymptote at x = 0, a hole in the graph at x = 2 and a horizontal asymptote at y = 1. Here is the graph with a few of the points that we have in our tables. Horizontal asymptote at y = 1. Vertical asymptote at x = 0. Hole in the graph.

12. Arrow Notation SymbolMeaning x  a  x approaches a from the right. x  a  x approaches a from the left. x   x approaches infinity; that is, x increases without bound. x    x approaches negative infinity; that is, x decreases without bound.

13. The line x  a is a vertical asymptote of the graph of a function f if f (x) increases or decreases without bound as x approaches a. f (x)   as x  a  f (x)   as x  a  The line x  a is a vertical asymptote of the graph of a function f if f (x) increases or decreases without bound as x approaches a. f (x)   as x  a  f (x)   as x  a  Thus, f (x)   or f(x)    as x approaches a from either the left or the right. f a y x x = a f a y x Definition of a Vertical Asymptote

14. The line x  a is a vertical asymptote of the graph of a function f if f (x) increases or decreases without bound as x approaches a. Thus, f (x)   or f(x)    as x approaches a from either the left or the right. x = a f a y x f a y x f (x)  as x  a  f (x)    as x  a  Definition of a Vertical Asymptote

15. Locating Vertical Asymptotes Ifis a rational function in which p(x) and q(x) have no common factors and a is a zero of q(x), the denominator, then x = a is a vertical asymptote of the graph of f.

16. The line y = b is a horizontal asymptote of the graph of a function f if f (x) approaches b as x increases or decreases without bound. f y x y = b x y f f y x f (x)  b as x   f (x)  b as x   f (x)  b as x  Definition of a Horizontal Asymptote

17. Locating Horizontal Asymptotes Let f be the rational function given by The degree of the numerator is n. The degree of the denominator is m. 1.If n<m, the x-axis, or y=0, is the horizontal asymptote of the graph of f. 2.If n=m, the line y = a n /b m is the horizontal asymptote of the graph of f. 3.If n>m,t he graph of f has no horizontal asymptote.

18. Strategy for Graphing a Rational Function Suppose that where p(x) and q(x) are polynomial functions with no common factors. 1. Find any vertical asymptote(s) by solving the equation q (x)  0. 2. Find the horizontal asymptote (if there is one) using the rule for determining the horizontal asymptote of a rational function. 3.Use the information obtained from the calculators graph and sketch the graph labeling the asymptopes.

19. Sketch the graph of

20. The vertical asymptote is x = -2 The horizontal asymptote is y = 2/5