1. Definition of a Rational Function A rational function is a quotient of polynomials that has the form The domain of a rational function consists of all real numbers except the zeroes of the polynomial in the denominator. S. Evans

2. Rational Functions and Their Characteristics (DAY 1): Over the next three days, you will examine the basic characteristics of a rational function. Understanding these functions will enable you to draw an accurate sketch of a rational function. These characteristics are: Domain Intercepts (both x and y) Holes in the function Asymptotes (Vertical, Horizontal, & Linear Oblique) Behaviour of the function close to the asymptote(s) Let’s start with familiar concepts, domain and intercepts….. S. Evans

3. Rational Functions and Their Characteristics (DAY 1): Example 1: Determine the domain and intercepts of each of the following: a) b) S. Evans

4. Rational Functions and Their Characteristics (DAY 1): The graph of a rational function usually has at least one asymptote, which may be vertical, horizontal, or oblique. An oblique asymptote is neither vertical nor horizontal. S. Evans.

5. Rational Functions and Their Characteristics (DAY 1): S. Evans Vertical Asymptotes (V.A.): Vertical asymptotes are imaginary vertical lines that form boundaries in the graph. Vertical asymptotes can be found where the function is undefined. However, not all restrictions produce a vertical asymptote. To make sure you have a vertical asymptote, simplify the rational function fully. The restrictions that are left in the reduced function will produce the vertical asymptotes. The equation of a vertical asymptote will then be. A graph never crosses a vertical asymptote because that x – value can never occur in the domain of the function.

6. Rational Functions and Their Characteristics (DAY 1): Holes: Sometimes a rational function has a hole in the graph. This is a point in the function where the graph gets really close to but never equals it. This will occur at the x – value of a restriction that is cancelled out during simplification. To get the y – value of the hole, substitute the x – value into the simplified rational function. Note: The x – value you are substituting in is a restriction and does not exist in the domain of the function. This is why the co-ordinates that you are finding are for a hole in the function. S. Evans.

7. Rational Functions and Their Characteristics (DAY 1): Example 2: Determine the vertical asymptotes and or holes of the following rational functions AND sketch on a Cartesian plane. a) S. Evans.

8. Rational Functions and Their Characteristics (DAY 1): S. Evans.

9. Rational Functions and Their Characteristics (DAY 1): b) S. Evans.

10. Rational Functions and Their Characteristics (DAY 1): S. Evans.

11. Rational Functions and Their Characteristics (DAY 1): Horizontal Asymptotes (H.A.): Horizontal asymptotes are imaginary horizontal lines that the graph will approach as x approaches very large positive values and/or very large negative values. S. Evans.

12. Rational Functions and Their Characteristics (DAY 1): Horizontal Asymptotes (H.A.): The following are general rules for finding a horizontal asymptote: Given that the numerator and denominator in the rational function are polynomials in of degree and, respectively. 1. If, then the horizontal asymptote is. 2. If, then the horizontal asymptote is. 3. If, there is no horizontal asymptote. S. Evans.

13. Rational Functions and Their Characteristics (DAY 1): Horizontal Asymptotes (H.A.): NOTE: Since a horizontal asymptote is only a barrier as x gets to be very large positive values and/or very large negative values, the graph may cross the horizontal asymptote in the central area of the graph. You will examine the behaviour of the curve in the next lesson. S. Evans.

14. Rational Functions and Their Characteristics (DAY 1): Example 3: Determine the equation of the horizontal asymptote, if it exists. a) S. Evans.

15. Rational Functions and Their Characteristics (DAY 1): b) c) S. Evans.

16. Rational Functions and Their Characteristics (DAY 1): Example 4: Given Find the domain, intercepts, and vertical and horizontal asymptotes. Then use this information to sketch what you now know about your function. S. Evans.

17. Rational Functions and Their Characteristics (DAY 1): Use this slide for work S. Evans.

18. Rational Functions and Their Characteristics (DAY 1): S. Evans.

19. Example 1: Determine the intercepts and the vertical and horizontal asymptotes for Then sketch what you know about the function so far. S. Evans

20. Rational Functions and Their Characteristics (DAY 2): S. Evans. Sketch what we know so far

21. Rational Functions and Their Characteristics (DAY 2): To be able to sketch the curve in the boundaries created by the asymptotes, you need to understand the behaviour of the graph as it approaches the asymptotes. Behaviour of the Function as it approaches the Vertical Asymptote (VA): (Does the curve point toward positive or negative infinity on either side of the VA?) You need to examine the sign of the function as x “approaches” the VA from the left side ( ) and the sign of the function as x “approaches” the VA from the right side ( ). To do this for the left side, you substitute a value smaller than the VA but really close to it and determine whether the output would be positive or negative. Then for the right side, you substitute a value larger than the VA but really close to it and determine whether the output would be positive or negative. A positive output would imply that the graph is pointing upwards towards positive infinity on that side of the VA and a negative output would imply that the graph is pointing downwards towards negative infinity on that side of the VA. S. Evans.

22. Rational Functions and Their Characteristics (DAY 2): Example 2: Examine the behaviour of the function (From ex 1) around the vertical asymptote. Then add that information to your sketch of S. Evans.

23. Rational Functions and Their Characteristics (DAY 2): Behaviour of the Function as it approaches the Horizontal Asymptote: (Is the function above or below the HA as x is a very large positive (as ) or as x is a very large negative (as )? When is the function above the HA? Solve: ___________________________ When is the function below the HA? Solve: ____________________________ Does the function cross the HA? Solve: ______________________________ As, is the function above or below the HA? You should substitute a very ________ _________ x – value and if the output is _____________________ the function is above the HA and if the output is _______________________ the function is below the HA. As is the function above or below the HA? You should substitute a very _________ ___________ x – value and if the output is _____________________ the function is above the HA and if the output is _______________________ the function is below the HA. S. Evans.

24. Rational Functions and Their Characteristics (DAY 2): Example 3: a) Does the function from ex 1) cross its horizontal asymptote? (Justify) S. Evans.

25. Rational Functions and Their Characteristics (DAY 2): c) As is the function,, above or below the HA? S. Evans.

26. Rational Functions and Their Characteristics (DAY 2): b) As is the function,, above or below the HA? S. Evans.

27. Rational Functions and Their Characteristics (DAY 2): d) Add the above information to your sketch of (from Ex 1) e) Given that curves usually “hug” the asymptotes, fill-in the possible sketch of the function. S. Evans.

28. Rational Functions and Their Characteristics (DAY 2): (Linear) Oblique Asymptotes: For rational functions, linear oblique asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. The equation of the linear oblique asymptote can be found by dividing the numerator by the denominator using long division. (or synthetic division where appropriate) S. Evans.

29. Rational Functions and Their Characteristics (DAY 2): Example 4: Let. a) Determine the oblique asymptote. S. Evans.

30. Rational Functions and Their Characteristics (DAY 2):. b) Determine the intercepts S. Evans.

31. Rational Functions and Their Characteristics (DAY 2):. b) Determine the intercepts S. Evans.

32. Rational Functions and Their Characteristics (DAY 2):. c) Determine the vertical asymptote(s) and examine the behaviour around the VA. S. Evans.

33. Rational Functions and Their Characteristics (DAY 2): S. Evans. d) Use your analysis to sketch the graph of.

34. Rational Functions and Their Characteristics (DAY 3): S. Evans. Example 1: Use domain, intercepts, asymptotes and behaviour to sketch the function

35. Rational Functions and Their Characteristics (DAY 3): S. Evans.

36. Rational Functions and Rates of Change (DAY 4): (work on the assigned worksheet) S. Evans. Concepts from the previous chapter are now revisited using rational functions: Average rate of change (use given range) Instantaneous rate of change (Check three following intervals that get close to the independent value)