Question

Given the graph, find the standard form of the equation for the parabola and identify the directrix.

Answer 1

A vertical parabola with vertex (1,-4) and focus (1,-2) is given. It is required to find the standard form of the equation for the parabola.

Recall that the equation of a Vertical Parabola with vertex (h,k) and focus (h,k+1/4a) is given as:

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

Compare the given vertex (1,-4) with the form (h,k). It follows that:

`h=1,k=-4`

Compare the given focus (1,-2) with the form (h,k+1/4a). It follows that:

`k+(1)/(4a)=-2`

Substitute k=-4 into the equation and find the value of a:

`\begin{gathered} -4+(1)/(4a)=-2 \\ \Rightarrow(1)/(4a)=-2+4 \\ \Rightarrow(1)/(4a)=2 \\ \Rightarrow4a=(1)/(2) \\ \Rightarrow a=(1)/(8) \end{gathered}`

Substitute k=-4, h=1, and a=1/8 into the equation of a vertical parabola:

`y=(1)/(8)(x-1)^2+(-4)`

Rewrite the equation in standard form as follows:

`\begin{gathered} y+4=(1)/(8)(x-1)^2 \\ \Rightarrow8(y+4)=(x-1)^2 \\ \text{ Swap the sides of the equation:} \\ \Rightarrow(x-1)^2=8(y+4) \end{gathered}`

Recall that the Directrix of a vertical parabola is represented by the equation:

`y=k-(1)/(4a)`

Substitute k=-4 and a=1/8 into the equation of directrix:

`\begin{gathered} y=-4-(1)/(4((1)/(8)))=-4-(1)/((1)/(2))=-4-2=-6 \\ \Rightarrow y=-6 \end{gathered}`

Hence, the correct answer is (x-1)²=8(y+4); y=-6.

The correct answer is (x-1)² = 8(y+4) ; y = - 6. (second option)