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 property of parabola

For a parabola \(x^{2} = 4py\), the vertex is \((0,0)\) in standard form. But if the parabola is \(x^{2}=4p(y - k)\), the vertex is \((0,k)\). Also, the focus of the parabola \(x^{2}=4p(y - k)\) is \((0,k + p)\).
From the graph, the vertex \((h,k)=(0,1)\).

Step2: Calculate the distance between vertex and focus

Looking at the graph, the distance between the vertex \((0,1)\) and the focus (by counting the units on the \(y -\)axis). The distance \(p = 1\) (since the vertex is at \(y = 1\) and the focus is at \(y=2\)).
Using the formula for the focus of the parabola \(x^{2}=4p(y - k)\) with \(h = 0,k = 1,p = 1\), the focus is \((0,k + p)=(0,1 + 1)\)

Answer:

The focus is \((0,2)\)