Questions: Given (f(x)=-3 x^2-18 x-22), write the function in vertex form. Write your answer in function notation by starting your answer with (f(x)=)

Solution Steps

To convert the quadratic function \( f(x) = -3x^2 - 18x - 22 \) into vertex form, we need to complete the square. The vertex form of a quadratic function is given by \( f(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola.

1. Factor out the coefficient of \( x^2 \) from the first two terms.
2. Complete the square inside the parentheses.
3. Adjust the constant term outside the parentheses to maintain equality.
4. Write the function in the form \( f(x) = a(x-h)^2 + k \).

We start with the quadratic function given by \[ f(x) = -3x^2 - 18x - 22. \]

We factor out the coefficient of \( x^2 \) from the first two terms: \[ f(x) = -3\left(x^2 + 6x\right) - 22. \]

To complete the square, we take the coefficient of \( x \) (which is 6), halve it to get 3, and then square it to obtain 9. We add and subtract this value inside the parentheses: \[ f(x) = -3\left(x^2 + 6x + 9 - 9\right) - 22. \] This simplifies to: \[ f(x) = -3\left((x + 3)^2 - 9\right) - 22. \]

Distributing the \(-3\) gives: \[ f(x) = -3(x + 3)^2 + 27 - 22. \] Thus, we have: \[ f(x) = -3(x + 3)^2 + 5. \]

Final Answer