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

Answer:

$y=-\frac{1}{8}(x-1)^2-2$

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

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

Now, the formula for the parabola is noted as:

$y=\frac{1}{4p}(x-h)^2+k$

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:

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

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

The equation is, therefore:

$y=-\frac{1}{8}(x-1)^2-2$