Answer:
Equation: [tex]f'"(x) = -\frac{1}{x-5}-7[/tex]
Asymptotes: [tex](x,y) = (5,-7)[/tex]
Step-by-step explanation:
Given
Let the reciprocal function be:
[tex]f(x) = \frac{1}{x}[/tex]
First, it was reflected across the x-axis.
The rule is: (x,-y)
So, we have:
[tex]f'(x) = -\frac{1}{x}[/tex]
Next, translated 5 units right.
The rule is: (x,y)=>(x-5,y)
So:
[tex]f"(x)= -\frac{1}{x-5}[/tex]
Lastly, translated 7 units down.
The rule is: (x,y) => (x,y-7)
So:
[tex]f'"(x) = -\frac{1}{x-5}-7[/tex]
To get the vertical asymptote, we simply equate the denominator to 0.
i.e.
[tex]x - 5 = 0[/tex]
[tex]x = 5[/tex]
To get the horizontal asymptote, we simply equate y to the constant.
i.e.
[tex]y = -7[/tex]
[tex](x,y) = (5,-7)[/tex]
