# 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...

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

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### 2 Answers

(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.)

**Sources:**

**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! :)