Question

1. determine the standard form for the equation of the parabola with vertex at (0, 0) and focus at (-2, 0).

$y^{2}=8x$
$y^{2}=-8x$
$x^{2}=-8y$
$y^{2}=-2x$

1. determine the standard form for the equation of the parabola with vertex at (3, 1) and focus at (3, 4).

$(x - 3)^{2}=12(y - 4)$
$(x + 3)^{2}=-12(y + 1)$
$(y - 1)^{2}=12(x - 3)$
$(x - 3)^{2}=12(y - 1)$

1. determine the standard form for the equation of the parabola with directrix at x = -2 and focus at (2, -4).

$x^{2}=8y$
$(y + 4)^{2}=-8x$
$(y + 4)^{2}=8x$
$y^{2}=8(x + 4)$

Explanation:

Step1: Recall parabola formula for horizontal axis with vertex at origin

The standard - form equation of a parabola with vertex \((h,k)=(0,0)\) and horizontal axis of symmetry is \(y^{2}=4px\), where the focus is \((p,0)\). Given the focus is \((- 2,0)\), then \(p=-2\).
Substitute \(p = - 2\) into \(y^{2}=4px\), we get \(y^{2}=4\times(-2)x=-8x\).

Step2: Recall parabola formula for vertical axis with non - origin vertex

The standard - form equation of a parabola with vertex \((h,k)=(3,1)\) and vertical axis of symmetry is \((x - h)^{2}=4p(y - k)\). The distance between the vertex \((3,1)\) and the focus \((3,4)\) is \(p=4 - 1=3\).
Substitute \(h = 3\), \(k = 1\), and \(p = 3\) into \((x - h)^{2}=4p(y - k)\), we have \((x - 3)^{2}=4\times3\times(y - 1)=(x - 3)^{2}=12(y - 1)\).

Step3: Find vertex and \(p\) for the third parabola

The vertex of a parabola is the mid - point between the focus \((2,-4)\) and the directrix \(x=-2\). The \(x\) - coordinate of the vertex is \(\frac{2+( - 2)}{2}=0\), and the \(y\) - coordinate of the vertex is \(-4\), so the vertex is \((0,-4)\). The distance between the vertex \((0,-4)\) and the focus \((2,-4)\) is \(p = 2\). Since the axis of symmetry is horizontal (because the directrix is a vertical line \(x=-2\)), the standard - form equation is \((y - k)^{2}=4p(x - h)\). Substitute \(h = 0\), \(k=-4\), and \(p = 2\) into the equation, we get \((y+4)^{2}=4\times2\times x=(y + 4)^{2}=8x\).

Answer:

1. \(y^{2}=-8x\)
2. \((x - 3)^{2}=12(y - 1)\)
3. \((y + 4)^{2}=8x\)