# Describe the asymptotes for the function `y=5/(x+3)-2`     25a) Define or explain what an asymptote is 25b) For the function y= 5/(x + 3) – 2 determine the vertical asymptote 25c) For the function y= 5/(x + 3) – 2 are the asymptotes right or left of those of the parent function, y=1/x 25d) For the function y= 5/(x + 3) – 2 are the asymptotes above or below those of the parent function y=1/x

(a) There are two types of asymptotes for this problem. A vertical asymptote is a vertical line near which the function grows without bound (gets infinitely large or small). If the vertical asymptote is at x=c, then the function is undefined at c.

There are different definitions for horizontal asymptotes -- they are horizontal lines that the function approaches as x grows without bound. Some authors put restrictions (e.g. the function cannot cross the asymptote, or cannot cross it infinitely many times,etc...).

(b) The vertical asymptote is at x=-3. You cannot divide by zero, thus the function is undefined at x=-3. As x approaches -3 from the left, the function decreases without bound and as x approaches -3 from the right the function increases without bound.

(c) For `y=1/x` , the vertical asymptote is at x=0. Thus the vertical asymptote for this function is 3 units to the left.

(d) For `y=1/x` , the horizontal asymptote is at y=0. Thus the horizontal asymptote is 2 units below that of its parent function.

** for (c) and (d) you have `y=1/x` as the parent function, and `y=a/(x-h)+k` as the transformed function. `a` makes a vertical stretch/compression (if a<0 it reflects the graph over the x-axis); `h` is a horizontal translation and `k` is a vertical translation**

The graph:(The asymptotes are drawn, but they are not part of the graph of the function. They are there as graphing aids.)