Question

Write the equation $y=-x^2-4x+6$ in the form $y=a{\left(x-h\right)}^2+k$.

Short answer

The required form of the equation in$y=a{\left(x-h\right)}^2+k$is $y=-{\left(x+2\right)}^2+10$ where $a=-1,h=-2\text{and}k=10$.

Step-by-step solution

Step 1. Write down the given information.

The given equation is $y=-x^2-4x+6$.

Step 2. Calculation.

The given equation $y=-x^2-4x+6$ is converted to$y=a{\left(x-h\right)}^2+k$form as:

$\begin{array}{c}y=-x^2-4x+6\\ =-\left(x^2+4x-6\right)\\ =-\left(x^2+4x-6+{\left(2\right)}^2-{\left(2\right)}^2\right)\mathrm{....}\left(\text{Addandsubtract}{\left(2\right)}^2\right)\\ =-\left(x^2+4x+{\left(2\right)}^2-\left(6+{\left(2\right)}^2\right)\right)\\ =-\left({\left(x+2\right)}^2-10\right)\\ =-{\left(x+2\right)}^2+10\end{array}$

Here $a=-1,h=-2\text{and}k=10$.

Step 3. Conclusion.

The required form of the given equation is $y=-{\left(x+2\right)}^2+10$ where $a=-1,h=-2\text{and}k=10$.