Question

a. vertex (4, -2), focus (2, -2)
equation:__________________________

Explanation:

Step1: Determine the form of the parabola

Since the \(y\) - coordinate of the vertex \((4,-2)\) and the focus \((2,-2)\) is the same, the parabola opens horizontally. The standard form of a horizontally - opening parabola is \((y - k)^{2}=4p(x - h)\), where \((h,k)\) is the vertex. Here, \(h = 4\), \(k=-2\).

Step2: Calculate the value of \(p\)

The distance between the vertex \((h,k)\) and the focus \((h + p,k)\) is given by \(p\). We know that \(h + p=2\), \(h = 4\). Substituting \(h\) into \(h + p=2\), we get \(4 + p=2\), so \(p=-2\).

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

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

Answer:

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