Exercise 46 Page 522

y=0.19x+210 000/x-500 To graph the rational function, we will follow three steps.

1. Find the horizontal and vertical asymptotes.
2. Plot some points around the vertical asymptote.
3. Sketch the graph.

   Finding the Asymptotes

   Since the degrees of the numerator and the denominator are the same, the horizontal asymptote is the ratio of the coefficient of the terms of greatest degree in the numerator and the denominator. y=0.19x+210 000/( 1)x-500 ⇓ y=0.19/1= 0.19 Therefore, the horizontal asymptote is y=0.19. We see that 500 is the zero of the denominator and not a zero of the numerator. This means that x= 500 is the vertical asymptote. y=0.19x+210 000/x-500 l → x= 500 Let's draw the asymptotes.

   

   Plotting Some Points

   Let's find some points both to the left and the right of the vertical asymptote.

   x	0.19x+210 000/x-500	y=0.19x+210 000/x-500
   Left of the Asymptote	- 300	0.19( - 300)+210 000/- 300-500	≈ - 262
   	0	0.19( 0)+210 000/0-500	- 420
   	300	0.19( 300)+210 000/300-500	≈ - 1050
   Right of the Asymptote	700	0.19( 700)+210 000/700-500	≈ 1050
   	1000	0.19( 1000)+210 000/1000-500	≈ 420
   	1300	0.19( 1300)+210 000/1300-500	≈ 262

   Let's plot the points (x, y) so we can see the behavior of the function.

   

   Sketching The Graph

   Finally, we will use the points to sketch the graph. It must approach both the horizontal and vertical asymptotes.