Problem

Write an equation in the form $y=a(x-p)^{2}+q$ for a parabola congruent to $y=-5 x^{2}$ with a minimum of -6 and an axis of symmetry of $x=2$.
$a=-5$

Answer

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Answer

\(\boxed{y = -5(x - 2)^2 - 6}\) is the final answer.

Steps

Step 1 :Given the equation of the parabola in the form \(y = ax^2\), we want to write it in the form \(y = a(x-p)^2 + q\).

Step 2 :Since the parabola is congruent to the given one, the value of \(a\) is the same in both equations, which is -5.

Step 3 :The minimum of the parabola is -6, which is the value of \(q\).

Step 4 :The axis of symmetry of the parabola is \(x = 2\), which is the value of \(p\).

Step 5 :Substitute these values into the equation \(y = a(x-p)^2 + q\) to get the equation of the parabola.

Step 6 :So, the equation of the parabola is \(y = -5(x - 2)^2 - 6\).

Step 7 :\(\boxed{y = -5(x - 2)^2 - 6}\) is the final answer.

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