Question

Write each equation in vertex form and find the vertex

Answer 1

The vertex form of a quadratic equation is the following:

$y=a(x-h)^2+k$

where the vertex is (h,k).

So, we need to convert the given standard form into a vertex form. Then, by factoring a number -3, we have

$y=-3(x^2+4x+\frac{7}{3})$

We can note that

$\begin{gathered} (x+2)^2=x^2+4x+4 \\ \text{then} \\ x^2+4x=(x+2)^2-4 \end{gathered}$

Therefore, we can rewrite our last result as

$y=-3((x+2)^2-4+\frac{7}{3})$

Since

$\begin{gathered} -4+\frac{7}{3}=\frac{7}{3}-4=\frac{7}{3}-\frac{12}{3} \\ \text{then} \\ -4+\frac{7}{3}=-\frac{5}{3} \end{gathered}$

So, we have obtained

$y=-3((x+2)^2-\frac{5}{3})$

Finally, by distributing the number -3 into the parentheses, the vertex form is given by:

$y=-3(x+2)^2+5$

Then, by comparing this result with the vertex form from above, the vertex (h,k) is

$(h,k)=(-2,5)$