Analyze the graph of the function R(x)=x^2+3x-54/x-9

Question

A). What is the domain of R(x)?

b) What is the equation of the vertical asymptote(s) of R(x)?

C) What is the equation of the horizontal or oblique asymptote of R(x)?

Answer 1

The rational function

The graph of rational functions can be recognized by the fact two or more parts.

• To find  y intercept  x = 0 in the rational function.

In this case y  intercept is 6.

• To find x intercept let the numarator = 0

In this case x  intercepts are - 9 and 6.

• Vertical asymptotes are found  by solving denominator = 0

x - 9 = 0

x = 9

Vertical asymptote is x  = 9.

• To find oblique asymptote.

Degree of the numarator = 2 and the degree of denominator = 1.

If the degree of the numerator is one grater than the degree of the denominator, then the function will have oblique asymptote.

oblique asymptote is found by long division.

Quotient is oblique asymptote.

In this case oblique asymptote is y  = x  + 12.

Answer 2

Contd...

Choose random values for x and find the corresponding values for y.

x	y = (x2 + 3x - 54) /(x - 9)	(x , y )
-20	y = [(-20)2 + 3(-20) - 54)] /(-20 - 9)	(-20, -9.86)
-30	y = [(-30)2 + 3(-30) - 54)] /(-30 - 9)	(-30 , -19.38)
20	y = [(20)2 + 3(20) - 54)] /(20 - 9)	(20, 36.9)
30	y = [(30)2 + 3(30) - 54)] /(30 - 9)	(30,44.57)

Graph

1) Draw the coordinate plane.

2) Next dash the oblique and vertical asymptotes.

3) Plot the x,y intercept and coordinate pairs found in the table..

4) Connect the plotted points .

When you draw your graph, use smooth curves complete the graph.

A)Domain of the function all real numbers except x = 9.

B)Vertical asymptote is x  = 9.

C)oblique asymptote is y  = x  + 12