Question

the graph to the right shows a parabola along with its vertex, focus, and directrix. determine the coordinates of the vertex and focus, and the equation of the directrix. what are the coordinates of the vertex? (-4, -1) (type an ordered pair.) what are the coordinates of the focus? (-4, -1.5) (type an ordered pair.) what is the equation of the directrix? = (simplify your answer.)

Explanation:

Step1: Recall the relationship between vertex, focus and directrix

For a parabola, the vertex is the mid - point between the focus and the directrix. The distance between the vertex \((h,k)\) and the focus \((h,k + p)\) is \(|p|\), and the equation of the directrix is \(y=k - p\) (for a parabola of the form \((x - h)^2=4p(y - k)\)).
We know the vertex \(V=(h,k)=(-4,-1)\) and the focus \(F=(h,k + p)=(-4,-1.5)\).

Step2: Calculate \(p\)

Since \(k=-1\) and \(k + p=-1.5\), then \(p=-1.5-(-1)=-0.5\)

Step3: Find the equation of the directrix

Using the formula \(y = k-p\), substitute \(k=-1\) and \(p = - 0.5\). Then \(y=-1-(-0.5)\)

Answer:

\(y=-0.5\)