Question

re-write the quadratic function below in standard form
$y = -2(x - 6)^2 - 3$

Explanation:

Step1: Expand the square term

First, we expand \((x - 6)^2\) using the formula \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = x\) and \(b = 6\), so \((x - 6)^2 = x^2 - 12x + 36\).
The function becomes \(y = -2(x^2 - 12x + 36) - 3\).

Step2: Distribute the -2

Multiply each term inside the parentheses by -2: \(-2\times x^2 = -2x^2\), \(-2\times(-12x) = 24x\), \(-2\times36 = -72\).
So now we have \(y = -2x^2 + 24x - 72 - 3\).

Step3: Combine like terms

Combine the constant terms -72 and -3: \(-72 - 3 = -75\).
The function in standard form (which is \(y = ax^2 + bx + c\)) is \(y = -2x^2 + 24x - 75\).

Answer:

\(y = -2x^2 + 24x - 75\)