1. Rational Functions Sec. 2.7a

2. Definition: Rational Functions Let f and g be polynomial functions with g (x ) = 0. Then the function given by is a rational function. The domain of a rational function is all reals except the zeros of its denominator. Every rational function is continuous on its domain.

3. Finding the Domain of a Rational Function Find the domain of the given function and use limits to describe its behavior at value(s) of x not in its domain. Domain? Now, sketch the graph… What does the function approach as x approaches 2?

4. A Reminder about Asymptotes The line y = b is a horizontal asymptote if or The line x = a is a vertical asymptote if Does this make sense with our previous example???

5. Now, Let’s Analyze the Reciprocal Function… Domain: Range: Continuity: Continuous on D Inc/Dec:Dec. on Symmetry: Origin (odd function) Boundedness:Unbounded Local Extrema:None H.A.: V.A.: End Behavior:

6. Transforming the Reciprocal Function Describe how the graph of the given function can be obtained by transforming the graph of the reciprocal function. Identify the horizontal and vertical asymptotes and use limits to describe the corresponding behavior. Sketch the graph of the function. Translate f (x) left 3 units, then vertically stretch by 2 H.A:

7. Transforming the Reciprocal Function Describe how the graph of the given function can be obtained by transforming the graph of the reciprocal function. Identify the horizontal and vertical asymptotes and use limits to describe the corresponding behavior. Sketch the graph of the function. Translate f (x) left 3 units, then vertically stretch by 2 V.A:

8. Transforming the Reciprocal Function Let’s do the same thing with a new function: Begin with polynomial division: Translate f (x) right 2, reflect across x-axis, translate up 3 H.A:

9. Transforming the Reciprocal Function Let’s do the same thing with a new function: Begin with polynomial division: Translate f (x) right 2, reflect across x-axis, translate up 3 V.A:

10. Find the horizontal and vertical asymptotes of the given function. Use limits to describe the corresponding behavior of the function. What’s the Domain?  So there are no vertical asymptotes!!! Why not??? First, let’s solve this algebraically… Now…Limits and Asymptotes of Rational Functions

11. Find the horizontal and vertical asymptotes of the given function. Use limits to describe the corresponding behavior of the function. First, let’s solve this algebraically… To find horizontal asymptotes, first use polynomial division: As x becomes very large or very small, this last term approaches zero… Why? So, the horizontal asymptote is the line y = 1 Using limit notation: Verify graphically?

12. The graphs of Graphs of Rational Functions have the following characteristics: 1. End Behavior Asymptote: If n < m, the end behavior asymptote is the horizontal asymptote of y = 0. If n = m, the end behavior asymptote is the horizontal asymptote. If n > m, the end behavior asymptote is the quotient polynomial function y = q(x), where f (x) = g(x)q(x) + r(x). There is no horizontal asymptote.

13. The graphs of Graphs of Rational Functions have the following characteristics: 2. Vertical Asymptotes: These occur at the zeros of the denominator, provided that the zeros are not also zeros of the numerator of equal or greater multiplicity. 3. x-intercepts: These occur at the zeros of the numerator, which are not also zeros of the denominator. 4. y-intercept: This is the value of f (0), if defined.

14. Find the asymptotes and intercepts of the given function, and then graph the function. Degree of Numerator > Degree of Denominator  long division! The quotient q(x) = x is our slant asymptote Factor the denominator: Vertical Asymptotes are at x = 3 and x = –3 x-intercept = 0, y-intercept = f (0) = 0 Verify all of this graphically???

15. Guided Practice Find the horizontal and vertical asymptotes of the given function. Use limits to describe the corresponding behavior. No vertical asymptotes H.A.: y = 3

16. Guided Practice Find the horizontal and vertical asymptotes of the given function. Use limits to describe the corresponding behavior. V.A.: x = 0, x = –3 H.A.: y = 0

17. Whiteboard Practice Find the asymptotes and intercepts of the given function, then graph the function. Intercepts: (0, –2/3), (–2, 0) Asymptotes: x = –3, x = 1, y = 0

18. Whiteboard Practice Find the asymptotes and intercepts of the given function, then graph the function. No Intercepts Asymptotes: x = –2, x = 0, x = 2, y = 0

19. Whiteboard Practice Find the asymptotes and intercepts of the given function, then graph the function. Intercepts: (0, –3), (–1.840, 0), (2.174, 0) Asymptotes: x = –2, x = 2, y = –3

20. Whiteboard Practice Find the asymptotes and intercepts of the given function, then graph the function. Intercepts: (0, –7/3), (–1.541, 0), (4.541, 0) Asymptotes: x = –3, y = x – 6