1. Concept

2. Graph with No Horizontal Asymptote
Example 1

3. 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.
Example 2A

4. Original equation r2 = 15 Simplify.
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.
Example 2A

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

6. 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.
Example 2B

7. 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.
Example 2C

8. Concept

9. x = 0 Take the square root of each side. There is a zero at x = 0.
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.
Example 3

10. Step 2 Find the asymptotes. x + 1 = 0 Set b(x) = 0.
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.
Example 3

11. The equation of the asymptote is the quotient excluding any remainder.
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.
Example 3

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