Question

write an equation in standard form for the parabola
a parabola with a vertex at (3,2) and a focus at (3,4)
$(x - 3)^2 = 8(y - 2)$
edit my response
a parabola with a vertex at (4,1) and a directrix at
$x = 5$
$(y - 1)^2 = - 4(x - 4)$
edit my response

Explanation:

Step1: Recall the standard form of a parabola

The standard form of a parabola that opens up or down is \((x - h)^{2}=4p(y - k)\), where \((h,k)\) is the vertex and \((h,k + p)\) is the focus.

Step2: Identify \(h\), \(k\), and \(p\)

Given the vertex \((h,k)=(3,2)\) and the focus \((h,k + p)=(3,4)\). So, \(h = 3\), \(k = 2\), and \(k + p=4\). Solving \(2 + p=4\) gives \(p = 2\).

Step3: Substitute into the standard form

Substitute \(h = 3\), \(k = 2\), and \(p = 2\) into \((x - h)^{2}=4p(y - k)\). We get \((x - 3)^{2}=4\times2(y - 2)\), which simplifies to \((x - 3)^{2}=8(y - 2)\).

Answer:

\((x - 3)^{2}=8(y - 2)\)