Question

the graph 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 is the vertex? (0,1) (type an ordered pair.) what is the focus? (0,7) (simplify your answer. type an ordered pair.) choose the correct equation of the directrix. a. ( y=\frac{1}{24}x ) b. ( x=-5 ) c. ( y=-5 ) d. ( x=\frac{1}{24}y )

Explanation:

Step1: Recall the property of parabola

For a parabola, the vertex is the mid - point between the focus and the directrix. The parabola is of the form \(x^{2}=4p(y - k)\) (opens up or down), where \((h,k)\) is the vertex. The distance between the vertex \((h,k)\) and the focus \((h,k + p)\) is \(p\), and the equation of the directrix is \(y=k - p\).
We know the vertex \((h,k)=(0,1)\) and the focus \((h,k + p)=(0,7)\).

Step2: Calculate \(p\)

Since \(k + p=7\) and \(k = 1\), then \(p=7 - 1=6\).

Step3: Find the equation of the directrix

Using the formula \(y=k - p\), substitute \(k = 1\) and \(p = 6\). So \(y=1-6=-5\).

Answer:

C. \(y = - 5\)