The given rational function is
![f(x) = \frac{x+2}{x-3}](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cfrac%7Bx%2B2%7D%7Bx-3%7D%20)
Part 1
The horizontal asymptote is obtained by either long division or synthetic division. It may be obtained also as
![f(x)= \frac{x-3+5}{x-3} = \frac{x-3}{x-3} + \frac{5}{x-3} =1+ \frac{5}{x-3}](https://tex.z-dn.net/?f=f%28x%29%3D%20%5Cfrac%7Bx-3%2B5%7D%7Bx-3%7D%20%3D%20%5Cfrac%7Bx-3%7D%7Bx-3%7D%20%2B%20%5Cfrac%7B5%7D%7Bx-3%7D%20%3D1%2B%20%5Cfrac%7B5%7D%7Bx-3%7D%20)
Therefore the horizontal asymptote is
y = 1.
The vertical asymptote occurs when the denominator is zero because the function becomes undefined. Set x-3 = 0 to obtain
x = 3.
Therefore a vertical asymptote occurs at x = 3.
The x-intercept occurs when f(x) = y = 0. Set f(x)=0 to obtain
![\frac{x+2}{x-3} =0](https://tex.z-dn.net/?f=%20%5Cfrac%7Bx%2B2%7D%7Bx-3%7D%20%3D0)
For x≠3, obtain
x+2=0 => x = -2
The x-intercept is x = -2.
The y-intercept occurs when x=0. Set x=0 in f(x) to obtain
![f(0)= \frac{0+2}{0-3} =- \frac{2}{3}](https://tex.z-dn.net/?f=f%280%29%3D%20%5Cfrac%7B0%2B2%7D%7B0-3%7D%20%3D-%20%5Cfrac%7B2%7D%7B3%7D%20)
The y-intercept is
y = -2/3
Part 2
The graph of the function is shown below. It identifies the horizontal and vertical asymptotes, the x-intercept, and the y-intercept.