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Rewrite each equation as y-k=a(x-h)^(2) or x-h=a(y-k)^(2). Find the vertex, focus, and directrix of the parabola. a y-3=(2-x)^(2)

User Def Avi
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Final answer:

The equation is rewritten in standard form, revealing that the parabola opens upwards and has a vertex at (2, 3). The focus is at (2, 3.25) and the directrix is the line y = 2.75.

Step-by-step explanation:

The given equation is y - 3 = (2 - x)^2. We can rewrite this in the standard form of a parabola by expressing it as y - k = a(x - h)^2. In this case, we see that the parabola opens upwards since the x-term is squared and the coefficient a is positive. To find the vertex, set the inside of the parenthesis equal to zero, (2 - x) = 0, so x equals 2, and since y is already isolated, the constant term on the y-side is 3. This gives us the vertex (h, k) = (2, 3).

Now, the focus and directrix of the parabola are determined by the value of a, which in this case is 1, since 1(x - 2)^2 = (x - 2)^2. The distance from the vertex to the focus and from the vertex to the directrix is given by 1/(4a). Since a = 1, this distance is 1/4. The focus will be at (2, 3 + 1/4), which is (2, 3.25), and the directrix will be the line y = k - 1/(4a), which is y = 3 - 1/4, or y = 2.75.

User IhtkaS
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