Rational Functions

1. Rational Functions Transformations and Graphing

2. Rational Function

3. Parent Function for Simple Rational Functions Parent function: hyperbola Shape: ______________ Parts are called: ______________ Domain: _______________________ Range: _______________________ branches All reals EXCEPT 0 or x ≠ 0 All reals EXCEPT 0 or y ≠ 0

4. Example • Identify vertical and horizontal asymptotes of the following rational functions. x = 0 Vertical Asymptote (V.A.) _______ Horizontal Asymptote(H. A.)________ Domain__________________ Range ___________________ y = 0 All Real except x can’t equal zero All Real except y can’t equal zero

5. Example • Identify vertical and horizontal asymptotes of the following rational functions. x = 0 Vertical Asymptote (V.A.) _______ Horizontal Asymptote(H. A.)________ Domain__________________ Range ___________________ y = 0 All Real except x can’t equal zero

6. Transformations • Forms

7. a = vertical stretch/compression = reflection over the x-axish = horizontal shifttranslates it right or left vertical asymptotek = vertical shifttranslates it up or down horizontal asymptote

8. Translating Rational Functions Three steps to graphing rational functions in the form of x = h y = k DRAW • _______the asymptotes of _______ and ______. • _______ the points to the left and to the right of the __________ asymptote. Make a table to help! • _______ the two ____________ of the hyperbola so that they pass through the plotted points and approach the asymptotes. PLOT vertical DRAW branches

9. Ex 1: Graph Simple Rational Functions Graph the function . Compare the graph with the parent function. Step 1: Graph the parent function. Step 2: Draw the asymptotes. x = 0 and y = 0 Step 3: Plot points on either side of the asymptotes. Look at the table on your calculator. Step 4: Connect your points! Compare and contrast the graphs.

10. Ex 2: Translate Simple Rational Functions Graph . State the domain and range. Step 1: Determinethe asymp- totes. Draw them in! Remember that x = h and y = k x = -2 and y = -1 Step 2: Plot points on either side of the asymptotes. Look at the table on your calculator. Calc: y = ( -4 / (x + 2)) -1 Step 3: Connect your points! All reals except x ≠ -2 Domain: ___________________________ Range: ____________________________ All reals except y ≠ -1

11. Example State the Domain, Range, the vertical and horizontal asymptote.

12. State the Domain, Range, the vertical and horizontal asymptote. Also, be able to state all the transformations that take place from the parent function. 1. 3. 2. 4.

13. Vertical Asymptotes Let be a rational function in lowest terms with the degree of at least 1 • Vertical (VA) • The graph of f(x) has a vertical asymptote corresponding to each root of . • Set each factor in the denominator equal to zero. Meaning you might have to factor, and most of the time will!!! ( denominator = 0) Solve for x. This will give the location of the vertical asymptote. • If there is not a real solution of h(x)=0, then there is no vertical asymptote.

14. Find the Vertical Asymptotes 1. 2. V.A. x = -3 V.A. x = 1 & x = -1 3. 4. V.A. x = 4 & x = -1 V.A.= none h(x)= no real solutions

15. Horizontal Asymptotes Let be a rational function in lowest terms with the degree of at least 1 • Numerator and denominator must be in standard form. • m and n are the degrees of the polynomials • Degree is the highest power of x when in standard form • The polynomial 2x3-4x+3 has a degree of _______. 3

16. Look at these equations and see what value of f(x) is approaching as it goes to infinity. 1. 2. 3. 4. 5. 6.

17. so y = ¾ Graphing General Rational Functions How do we find the HORIZONTAL ASYMPTOTE(S)? To do this, you will have to look at the highest powers in your numerator and denominator. You have three options: If the numerator power is bigger, then there is ________________________! NO HORIZONTAL asymptote! If the denominator power is bigger, then the ___________________________! HORIZONTAL asymptote is y = 0 If the powers are the SAME, then the asymptote is ____________________________! the RATIO of the coefficients

18. Asymptotes (HA-3 Cases) • Horizontal (Case 1) 1. m and n are the degrees of the polynomials • If the degree of g(x) is less than the degree of h(x), the x-axis is a horizontal asymptote. • If m < n, then the graph has a horizontal asymptote at y = 0.

19. Asymptotes (HA-3 Cases) • Horizontal (Case 2) 2. If the degree of g(x) equals the degree of h(x), then the horizontal asymptote is determined by the ratio of the leading coefficients. • If m = n, then the graph of the function has a horizontal asymptote at .

20. Asymptotes (HA-3 Cases) • Horizontal (Case 3) 3. m > n If the degree of g(x) is greater than the degree of h(x), then there is no horizontal asymptote.

21. BOBO BOTN • Bigger on Bottom the asymptote is at y = 0 • Bigger on Top there is no asymptote! • The third case occurs when the degrees are the same. In this case the asymptote is at the ratio of the coefficients.

22. Review BOBO: HA is at y = 0 • If m < n then _____________________ • If m > n then _____________________ • If m = n then _____________________ BOTN: No HA HA is at the ratio

23. Rational Functions—HOLE(S) With rational functions, there are sometimes when a hole (or several) will appear in the graph. How do we find these holes??? Hopefully you remember how to factor, because it is a big part. Directions for finding holes. Factor the numerator AND denominator if possible. Leave them in factored form. If there are factors that are the same, then set it equal to 0 and solve for x.

24. How do we find the HOLE(S)? Factor the numerator AND denominator if possible. Leave them in factored form. If there are factors that are the same, then set it equal to 0 and solve for x. Example: Factor the numerator Put your factors back into the fraction. Are there any factors that are the same? YES! (x + 1) Set the factor equal to 0 and solve. This will give you the hole!

25. Factor and Simplify • Factor the numerator and denominator & Simplify Hole x = -2 Hole x = 1 Hole x = -1

26. Factor and Simplify No • Enter the function into y1 and the answer into y2 • Are they two the same? We have a hole!

27. Putting it All Together… For the following equation, find the vertical and horizontal asymptotes and the holes, if they exist. Example 1: Find the vertical asymptote(s): Since the power on the top is bigger, there is no horizontal asymptote! Find the horizontal asymptote(s): Find the hole(s): There are no holes since we cannot “cancel” out any factors!

28. Putting it All Together… For the following equation, find the vertical and horizontal asymptotes and the holes, if they exist. Example 2: Find the vertical asymptote(s): Since the power on the top is bigger, there is no horizontal asymptote so it is a LINE! Find the horizontal asymptote(s): Find the hole(s): Since the common factor is (x - 3), there is a hole when x = 3

29. Putting it All Together… For the following equation, find the vertical and horizontal asymptotes and the holes, if they exist. Example 2: WAIT! Can you have an asymptote AND a hole at the same value? In this case, when x = 3??? NO!!! When you graph the function, what does it look like? a line Lines do NOT have asymptotes! Therefore, there is a hole when x = 3. SO…if you “cancel” out the common factors and a hole and asymptote exist at the same value, check to see if your equation reduces to a line!

30. Looking at the Symmetry… • Odd-Rotational about the origin. • Even-Symmetric about the y-axis.

31. Example • Odd or Even? Odd Even Odd

32. Practice the Parent Functions • Graph the following on your graphing calculator and determine whether each is Even, Odd or Neither. Neither Even Odd Neither Even Odd Odd Even Even Neither Odd Neither