1. Graphing Rational Functions Day 2

2. Concept

3. Graph with No Horizontal Asymptote
Example 1
Graph with No Horizontal Asymptote
Step 1 Find the zeros.
x3 = 0 Set a(x) = 0.
x = 0 Take the cube root of each side.
There is a zero at x = 0.

4. Graph with No Horizontal Asymptote
Example 1
Graph with No Horizontal Asymptote
Step 2 Draw the asymptotes.
Find the vertical asymptote.
x + 1 = 0 Set b(x) = 0.
x = –1 Subtract 1 from each side.
There is a vertical asymptote at x = –1.
The degree of the numerator is greater than the degree of the denominator. So, there is no horizontal asymptote.

5. Graph with No Horizontal Asymptote
Example 1
Graph with No Horizontal Asymptote
Step 3 Draw the graph.
Use a table to find ordered pairs on the graph. Then connect the points.

6. Graph with No Horizontal Asymptote
Example 1
Graph with No Horizontal Asymptote
Answer:

7. Your Turn Example 1!! B.
C. D. A B C D

8. Use Graphs of Rational Functions
Example 2A
Use Graphs of Rational Functions
A. AVERAGE SPEED A boat traveled upstream at r1 miles per hour. During the return trip to its original starting point, the boat traveled at r2 miles per hour. The average speed for the entire trip R is given by the formula Draw the graph if r2 = 15 miles per hour.

9. Use Graphs of Rational Functions
Example 2A
Use Graphs of Rational Functions
Original equation
r2 = 15
Simplify.
The vertical asymptote is r1 = –15. Graph the vertical asymptote and function. Notice the horizontal asymptote is R = 30.

10. Use Graphs of Rational Functions
Example 2A
Use Graphs of Rational Functions
Answer:

11. Example 2B B. What is the R-intercept of the graph?
Use Graphs of Rational Functions
B. What is the R-intercept of the graph?
Answer: The R-intercept is 0.

12. Use Graphs of Rational Functions
Example 2C
Use Graphs of Rational Functions
C. What domain and range values are meaningful in the context of the problem?
Answer: Values of r1 greater than or equal to 0 and values of R between 0 and 30 are meaningful.

13. Your Turn Example 2!!
Use this information to answer questions on the next 3 slides.
A. TRANSPORTATION A train travels at one velocity V1 for a given amount of time t1 and then another velocity V2 for a different amount of time t2.
The average velocity is given by Let t1 be the independent variable and let V be the dependent variable. Which graph is represented if V1 = 60 miles per hour, V2 = 30 miles per hour, and t2 = 1 hour?

14. Example 2
A. B.
C. D.

15. B. What is the V-intercept of the graph?
Example 2
B. What is the V-intercept of the graph?
A. 30
B. 1
C. –1
D. –30

16. Example 2
C. What values of t1 and V are meaningful in the context of the problem?
A. t1 is negative and V is between 30 and 60.
B. t1 is positive and V is between 30 and 60.
C. t1 is positive and V is is less than 60.
D. t1 and V are positive.

17. Concept

18. Determine Oblique Asymptotes
Example 3
Determine Oblique Asymptotes
Graph
Step 1 Find the zeros.
x2 = 0 Set a(x) = 0.
x = 0 Take the square root of each side.
There is a zero at x = 0.

19. Determine Oblique Asymptotes
Example 3
Determine Oblique Asymptotes
Step 2 Find the asymptotes.
x + 1 = 0 Set b(x) = 0.
x = –1 Subtract 1 from each side.
There is a vertical asymptote at x = –1.
The degree of the numerator is greater than the degree of the denominator, so there is no horizontal asymptote.
The difference between the degree of the numerator and the degree of the denominator is 1, so there is an oblique asymptote.

20. Determine Oblique Asymptotes
Example 3
Determine Oblique Asymptotes
Divide the numerator by the denominator to determine the equation of the oblique asymptote.
– 1
(–) 1
The equation of the asymptote is the quotient excluding any remainder.
Thus, the oblique asymptote is the line y = x – 1.

21. Determine Oblique Asymptotes
Example 3
Determine Oblique Asymptotes
Step 3 Draw the asymptotes, and then use a table
of values to graph the function.
Answer:

22. Your Turn Example 3!!
Graph B.
C. D.

23. Concept

24. Graph with Point Discontinuity
Example 4
Graph with Point Discontinuity
Graph
Notice that or x + 2. Therefore, the graph of is the graph of f(x) = x + 2 with a hole at x = 2.

25. Graph with Point Discontinuity
Example 4
Graph with Point Discontinuity
Answer:

26. Your Turn Example 4!! See ya in CLASS!!!!! 
Which graph below is the graph of ?
A. B.
C. D.