Concept. Example 1 Graph with No Horizontal Asymptote.

Graph with No Horizontal Asymptote

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.

Use Graphs of Rational Functions

Simplify.

The vertical asymptote is r1 = –15. Graph the vertical asymptote and function. Notice the horizontal asymptote is R = 30.

Original equation

r2 = 15

Use Graphs of Rational Functions

Answer:

Use Graphs of Rational Functions

B. What is the R-intercept of the graph?

Answer: The R-intercept is 0.

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.

Determine Oblique Asymptotes

Step 1 Find the zeros.

x2 = 0 Set a(x) = 0.

x = 0 Take the square root ofeach side.

There is a zero at x = 0.

Graph

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 isno horizontal asymptote.The difference between the degree of thenumerator and the degree of thedenominator is 1, so there is an obliqueasymptote.

Determine Oblique Asymptotes

Divide the numerator by the denominator to determine the equation of the oblique asymptote.

The equation of the asymptote is the quotientexcluding any remainder.

Thus, the oblique asymptote is the line y = x – 1.

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– 1

1

Determine Oblique Asymptotes

Answer:

Step 3 Draw the asymptotes, and then use a table of

values to graph the function.