Question

Write the equation $y=\frac{1}{2}{x}^2+12x-8$ in standard form.

Short answer

The required form of the equation in standard form$y=a{\left(x-h\right)}^2+k$ is $y=\frac{1}{2}{\left(x+12\right)}^2-80$ where $a=\frac{1}{2},h=-12\text{and}k=-80$.

Step-by-step solution

Step 1. Write down the given information.

The given equation is $y=\frac{1}{2}{x}^2+12x-8$.

Step 2. Calculation.

The given equation $y=\frac{1}{2}{x}^2+12x-8$ is converted to standard form $y=a{\left(x-h\right)}^2+k$ as:

$\begin{array}{c}y=\frac{1}{2}{x}^2+12x-8\\ =\frac{1}{2}\left(x^2+24x-16\right)\\ =\frac{1}{2}\left(x^2+24x-16+{\left(12\right)}^2-{\left(12\right)}^2\right)\mathrm{....}\left(\begin{array}{l}\text{Addand}\\ \text{subtract}{\left(12\right)}^2\end{array}\right)\\ =\frac{1}{2}\left(\left(x^2+24x+{\left(12\right)}^2\right)-\left(16+{\left(12\right)}^2\right)\right)\\ =\frac{1}{2}{\left(x+12\right)}^2-80\end{array}$

Here $a=\frac{1}{2},h=-12\text{and}k=-80$.

Step 3. Conclusion.

The required form of the given equation is$y=\frac{1}{2}{\left(x+12\right)}^2-80$ where $a=\frac{1}{2},h=-12\text{and}k=-80$.