Question

find the equation of the parabola with the following properties. express your answer in standard form.
symmetric with respect to the line ( y = - 3 )
directrix is the line ( x = - 1 )
( p = - 1 )

Explanation:

Step1: Determine the form of the parabola

Since the parabola is symmetric with respect to the line \(y = - 3\) (a horizontal line), it is a parabola of the form \((y - k)^{2}=4p(x - h)\)

Step2: Find the vertex \((h,k)\)

The directrix of the parabola \((y - k)^{2}=4p(x - h)\) is \(x=h - p\). Given \(p=-1\) and directrix \(x = - 1\).
Substitute into \(x=h - p\), we have \(-1=h-(-1)\), so \(h=-2\)
Since the parabola is symmetric about \(y=-3\), \(k = - 3\)

Step3: Substitute \(h,k,p\) into the standard form

Substitute \(h=-2,k = - 3,p=-1\) into \((y - k)^{2}=4p(x - h)\)
\((y+3)^{2}=4\times(-1)(x + 2)\)

Answer:

\((y + 3)^{2}=-4(x + 2)\)