Question

write the equation of the conic section shown below.

Explanation:

Step1: Determine the vertex form of a parabola

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

Step2: Find the value of \(p\)

The focus of a parabola \((y - k)^2=4p(x - h)\) is \((h + p,k)\). The focus is at \((-6,3)\). So \(h + p=-6\), substituting \(h=-4\) gives \(-4 + p=-6\), then \(p=-2\).

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

Substitute \(h=-4\), \(k = 3\), and \(p=-2\) into \((y - k)^2=4p(x - h)\). We get \((y - 3)^2=4\times(-2)(x + 4)\).

Answer:

\((y - 3)^2=-8(x + 4)\)