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? (simplify your answer. type an ordered pair.)

Explanation:

Step1: Recall the definition of a parabola's focus

The focus of a parabola is a fixed point. For a parabola \(y = ax^{2}+bx + c\), if the vertex is \((h,k)\) and the parabola opens upwards or downwards, the distance from the vertex to the focus is \(p\). In the standard form \((x - h)^{2}=4p(y - k)\), if \(p>0\) the parabola opens upwards, if \(p < 0\) it opens downwards.

Step2: Calculate the distance from the vertex to the focus

Looking at the graph, the vertex is \((0,1)\). The focus is 6 units above the vertex. So \(p = 6\).

Step3: Find the coordinates of the focus

Since the vertex is \((h,k)=(0,1)\) and the parabola opens upwards (\(p>0\)), using the formula for the focus \((h,k + p)\). Substitute \(h = 0\), \(k = 1\), \(p=6\) into the formula.
The focus is \((0,1 + 6)\)

Answer:

\((0,7)\)