Question

1. find the equation of the parabola with the focus at (-3,1) and vertex at (-1,1).

Explanation:

Step1: Determine the form of the parabola

Since the \(y -\)coordinates of the focus \((-3,1)\) and vertex \((-1,1)\) are 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.

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

Given the vertex \((h,k)=(-1,1)\). The distance between the vertex \((-1,1)\) and the focus \((-3,1)\) is \(p\). Using the distance formula for horizontal distance \(p=x_{focus}-x_{vertex}\), so \(p=-3-(-1)=-2\)

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

Substitute \(h = - 1,k = 1,p=-2\) into \((y - k)^{2}=4p(x - h)\)

$$
LATEXBLOCK0
$$

Answer:

\((y - 1)^{2}=-8(x + 1)\)