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+(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+(7)/(3))`

Since

`\begin{gathered} -4+(7)/(3)=(7)/(3)-4=(7)/(3)-(12)/(3) \\ \text{then} \\ -4+(7)/(3)=-(5)/(3) \end{gathered}`

So, we have obtained

`y=-3((x+2)^2-(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)`