If $(-24, 12)$ is a point on the graph of the function $y=f(x)$, identify one point on the graph of function $y = -2 \sqrt{f( -(x - 2)) - 4} + 6$.

Question

I am fairly confident I know the process to solving this equation. But the answer I am given in my textbook for the new $y$ value confuses me. Here is a brief overview my steps:

$(-24, 12)$ is the original point. $y= -2\sqrt{f(-(x - 2)) - 4} + 6$ is the new function the original point must be manipulated through.

1. I found that after distributing the negative, $x$ must be subtracted by $2$ and divided by $-1$. The answer I got was $26$ for $x$. The answer I am given is the same.

2. For $y$, I concluded that I need to subtract by $4$, take the square root, multiply by $-2$, then add $6$. The final answer for $y$ I got was $6 + 4\sqrt{2}$. The textbook says the answer is $6 + 4\sqrt{3}$.

Where did I make a mistake? I feel somewhere I confused the steps but I cannot find where. Thanks in advance for taking a look at my question.

Answer 1

It seems that both you and your textbook are a bit off concerning $y$.

Given is $(x_0,f(x_0)) = (\color{blue}{-24},\color{green}{12})$ and $y = -2\sqrt{f(-(x-2))-4} + 6$

It might be useful to set up equations and to solve them instead of "verbalizing" arithmetic operations:

• For $x$: $-(x-2) = \color{blue}{-24} \Leftrightarrow x= 26$
• For $y$: $y = -2\sqrt{f(\color{blue}{-24})-4} + 6 = -2\sqrt{\color{green}{12}-4} + 6 = 6 - 4\sqrt{2}$.