Question

Find the equation of the parabola with the following properties. Express your answer in standard fornFocus at (1, -4)Directrix is the x-axisAnswer 2 Points

Answer 1

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

First, to know the opening of the parabola, let us solve for p:

`\begin{gathered} \text{ Focus:}(1,-4) \\ \text{ Directrix: y}=0 \\ p=(0-(-4))/(2)=(4)/(2)=2 \end{gathered}`

Now, the formula for the parabola is noted as:

`y=(1)/(4p)(x-h)^2+k`

Since our p is 2, the vertex of the parabola would be at:

`\begin{gathered} v(1,-4+2) \\ v(1,-2) \end{gathered}`

This will now be our (h,k).

With these, we know that

h = 1

k = -2

p = 2

We substitute these values to the equation:

`\begin{gathered} y=(1)/(4p)(x-h)^(2)+k \\ y=(1)/(4(2))(x-1)^2+(-2) \\ y=(1)/(8)(x-1)^2-2 \end{gathered}`

Since our parabola is opening downward, we will add a negative sign in front of the equation.

The equation is, therefore:

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