Question

complete the square to re-write the quadratic function in vertex form: $y = -3x^2 - 18x - 19$

Explanation:

Step1: Factor out the coefficient of \(x^2\) from the first two terms

First, we factor out \(-3\) from the terms involving \(x\):
\(y = -3(x^2 + 6x) - 19\)

Step2: Complete the square inside the parentheses

To complete the square for \(x^2 + 6x\), we take half of the coefficient of \(x\) (which is \(6\)), square it, and add and subtract it inside the parentheses. Half of \(6\) is \(3\), and \(3^2 = 9\). So we have:
\(y = -3(x^2 + 6x + 9 - 9) - 19\)

Step3: Rewrite the perfect square trinomial

We can rewrite \(x^2 + 6x + 9\) as \((x + 3)^2\):
\(y = -3((x + 3)^2 - 9) - 19\)

Step4: Distribute the \(-3\)

Now we distribute the \(-3\) across the terms inside the parentheses:
\(y = -3(x + 3)^2 + 27 - 19\)

Step5: Combine the constant terms

Finally, we combine the constant terms \(27\) and \(-19\):
\(y = -3(x + 3)^2 + 8\)

Answer:

\(y = -3(x + 3)^2 + 8\)