SOLUTION: 13- Show work, graph the function using transformations, state domain, range and asymptotes F(x)=5- 1/(x+4)²

Algebra ->  Graphs -> SOLUTION: 13- Show work, graph the function using transformations, state domain, range and asymptotes F(x)=5- 1/(x+4)²       Log On


   

Question 669204: 13- Show work, graph the function using transformations, state domain, range and asymptotes
F(x)=5- 1/(x+4)²
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
F%28x%29=5-1%2F%28x%2B4%29%5E2 is related to the functions f%28x%29=1%2Fx and g%28x%29=1%2Fx%5E2.
You know that the function f%28x%29=1%2Fx has a graph like this:
graph%28300%2C300%2C-5%2C5%2C-5%2C5%2C1%2Fx%29.
The x-axis (y=0) is a horizontal asymptote that the function f%28x%29=1%2Fx approaches as x and -x become larger and larger,
and you know that the function
does not exist for x=0 because you cannot divide by zero.
Instead, you have a vertical asymptote at x=0,
and there is no limit to the values f%28x%29 can take.
It is never zero, but can take any other value.

Something similar happens with the function g%28x%29=1%2Fx%5E2,
which does not exist for x=0 and graphs as graph%28300%2C300%2C-5%2C5%2C-5%2C5%2C1%2Fx%5E2%29
The only difference is that g%28x%29=1%2Fx%5E2 is never negative.
The domain is all real numbers except zero; the range is all y%3E0, and the asymptotes are the same.

The function h%28x%29=1%2F%28x%2B4%29%5E2 is like g%28x%29=1%2Fx%5E2,
but everything is shifted 4 units to the left.
What happened to g%28x%29=1%2Fx%5E2 for x=0, x=1, and x
now happens for h%28x%29=1%2F%28x%2B4%29%5E2 at x=-4, x=-3, and x-4.
The graph for h%28x%29=1%2F%28x%2B4%29%5E2 is graph%28300%2C300%2C-9%2C1%2C-5%2C5%2C1%2F%28x%2B4%29%5E2%29.
The vertical asymptote is x=-4%29%2C%0D%0Aand+the+function+does+not+exist+for+%7B%7B%7Bx=-4.
(All other real numbers are in the domain of f%28x%29).

Adding a minus sign in front just flips the graph.
Everything becomes the reflection under the x-axis:
graph%28300%2C300%2C-9%2C1%2C-5%2C5%2C-1%2F%28x%2B4%29%5E2%29

Adding 5 to that, just moves the graph (and asymptotes) up by 5 units.
graph%28300%2C300%2C-9%2C1%2C-3%2C7%2C5-1%2F%28x%2B4%29%5E2%2C5%29
The domain is all the real numbers except x=-4.
The vertical asymptote is x=-4.
The range is all y%3C5.
The horizontal asymptote (shown in green) is y=5.