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.)
25a) An asymptote is a restriction on an x-value or a y-value.
Vertical asymptote = restriction on an x-value
Horizontal asymptote = restriction on a y-value
For example, if the vertical asymptote is x=5, that means that the function can never have an x value equal to 5.
25b) The vertical asymptote is x=-3.
This is because the denominator can never be equal to 0, therefore x cannot equal -3.
25c) The asymptotes are left of those of the parent function because the horizontal shift is 3 units to the left.
25d) The asymptotes are below those of the parent function because the vertical shift is 2 units down.
Hope this helped and good luck! :)