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Section 4-2 : Parabolas

10. Convert the following equations into the form \(y = a{\left( {x - h} \right)^2} + k\).

\[f\left( x \right) = - {x^2} - 8x - 18\]

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Start Solution

We’ll need to do the modified completing the square process described in the notes for this section.

The first step in this process is to make sure that we have a coefficient of one on the \({x^2}\). So, for this problem that means we need to factor a minus sign out of the quadratic to get,

\[f\left( x \right) = - \left( {{x^2} + 8x + 18} \right)\] Show Step 2

Next, we need to take one-half the coefficient of the \(x\), square it and then add and subtract it onto the equation.

\[{\left( {\frac{8}{2}} \right)^2} = {\left( 4 \right)^2} = 16\] \[\require{color}f\left( x \right) = - \left( {{x^2} + 8x \,{\color{Red} + 16 - 16} + 18} \right)\]

Make sure to do the adding/subtracting inside the parenthesis. If we did it outside of the parenthesis we would not be able to do the next step!

Show Step 3

Next, we need to factor the first three terms and combine the last two numbers to get,

\[f\left( x \right) = - \left( {{{\left( {x + 4} \right)}^2} + 2} \right)\] Show Step 4

Finally, all we need to do is multiply the 6 back through the parenthesis to get,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{f\left( x \right) = - {{\left( {x + 4} \right)}^2} - 2}}\]