Question

Can I get help writing an equation in standard form for the parabola

Answer 1

To solve this problem, first, we use the following form of the equation of a vertical parabola:

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

where (h,k) are the coordinates of the vertex.

Using the above form and the graph, we get:

`y=a(x+4)^2+2.`

Now, to determine the value of a, we consider the point (-2,0) on the parabola:

`0=a(-2+4)^2+2.`

Solving for a, we get:

`\begin{gathered} -2=a2^2, \\ -(2)/(4)=a, \\ a=-(1)/(2). \end{gathered}`

Therefore:

`y=-(1)/(2)(x+4)^2+2.`

Finally, we take the above equation to its standard form:

`\begin{gathered} y=-(1)/(2)(x^2+8x+16)+2, \\ y=-(1)/(2)x^2-4x-8+2, \\ y=-(1)/(2)x^2-4x-6. \end{gathered}`

Answer:

`y=-(1)/(2)x^2-4x-6.`