Determining the equation of $g(x)$ given intersecting points via another function?

Question

Here's the problem:

I know this is a system of quadratic-quadratic equations. My attempt was to convert the equation of $f(x)$ into the form $f(x) = a(x - p)^2 + q$, making the $a$ negative to determine the equation of $g(x)$.

$f(x) = (x^2 - 6x) +5$

$f(x) = (x^2 - 6x + (- \frac{6}{2})^2 - (- \frac{6}{2})^2) + 5$

$f(x) = (x^2 - 6x + 9) - 9 + 5$

$f(x) = (x - 3)^2 - 4$

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$f(x) = -(x - 3)^2 - 4$

I then plugged in the points into the equation, but they weren't solutions so I know I approached this problem wrong.

The textbook answer is $g(x) = -(x - 6)^2 + 13$, how should I get this vertex?

Answer 1

You were on the right track when you made the $a$ negative. Since the question says $f(x)$ is congruent to $g(x)$, you know that the $a$'s in both equations will only differ in sign.

As for $p$ and $q$, you have two points that $g(x)$ passes through and two unknowns in the equation. Looks like a system of equations should do the trick. Can you take it from here?