Question

find the equation for the parabola with the following vertex, focus, and directrix. vertex: (3, -3) focus: (5, -3) directrix: x = 1 (y - ?)² = (x - )

Explanation:

Step1: Recall the standard form of a parabola

The standard form of a parabola that opens to the right or left is \((y - k)^{2}=4p(x - h)\), where \((h,k)\) is the vertex.
Given vertex \((h,k)=(3,-3)\), so \(h = 3\) and \(k=-3\).

Step2: Calculate the value of \(p\)

The distance between the vertex \((h,k)\) and the focus \((h + p,k)\) (for parabola opening to the right).
We know \(h=3\), focus \((h + p,k)=(5,-3)\).
So \(h + p=5\), substituting \(h = 3\) gives \(3 + p=5\), then \(p=2\).

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

Substitute \(h = 3\), \(k=-3\), \(p = 2\) into \((y - k)^{2}=4p(x - h)\)
We get \((y-(-3))^{2}=4\times2(x - 3)\)

Answer:

\((y-(-3))^{2}=8(x - 3)\)
So the blanks are filled as \((y-\boldsymbol{-3})^{2}=\boldsymbol{8}(x-\boldsymbol{3})\)