I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
Section 4.2 : Parabolas
10. Convert the following equations into the form \(y = a{\left( {x - h} \right)^2} + k\).
\[f\left( x \right) = - {x^2} - 8x - 18\]Show All Steps Hide All Steps
Start SolutionWe’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 2Next, 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 3Next, 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 4Finally, all we need to do is multiply the 6 back through the parenthesis to get,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{f\left( x \right) = - {{\left( {x + 4} \right)}^2} - 2}}\]