Question

find the equation of the parabola with the following properties. express your answer in standard form. vertex at (-2, -3) focus at (-2, 3)

Explanation:

Step1: Determine the form of the parabola

Since the \(x -\)coordinates of the vertex \((-2,-3)\) and focus \((-2,3)\) are the same, the parabola is of the form \((x - h)^{2}=4p(y - k)\), where \((h,k)\) is the vertex. Here \(h=-2,k = - 3\).

Step2: Calculate the value of \(p\)

The distance between the vertex \((h,k)=(-2,-3)\) and the focus \((h,k + p)=(-2,3)\) is given by \(p\). Using the formula \(k + p=3\), substituting \(k=-3\), we get \(-3 + p=3\), so \(p = 6\).

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

Substitute \(h=-2,k=-3,p = 6\) into \((x - h)^{2}=4p(y - k)\). We have \((x+2)^{2}=4\times6(y + 3)\)

Step4: Simplify the equation

Expand the right - hand side: \((x + 2)^{2}=24(y + 3)\)

Answer:

\((x + 2)^{2}=24(y + 3)\)