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Chapter 6: Problem 56

Find the standard form of the equation of the parabola with the given characteristics. Vertex: (-1,2)\(;\) focus: (-1,0)

Short Answer

Expert verified
The standard form of the equation of the parabola is \(y=-2*(x+1)^2 + 2\)

Step by step solution

01

Determine the Vertex and the Focus

In this problem, the given vertex of the parabola is (-1,2) and the focus is (-1,0). The vertex form of a parabola is \(y=a(x-h)^2 + k\), where (h,k) are the coordinates of the vertex.
02

Determine the Direction of the Parabola

We can tell that the parabola opens downwards because the focus point is below the vertex.
03

Find the Value of 'a'

Let's calculate the distance between the vertex and the focus which is also the value of 'a' in the standard form. The formula to calculate the distance between two points ((x1,y1), (x2,y2)) on a coordinate plane is \(\sqrt{(x2-x1)^2+(y2-y1)^2}\). But since the parabola opens downwards, a is negative. Hence, a=-2.
04

Substitute the Values into the Vertex Form of the Equation

Substitute h=-1, k=2 and a=-2 into the vertex form of a parabola which gives the equation \(y=-2*(x+1)^2+2\).

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