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Chapter 6: Problem 53

Find the standard form of the equation of the parabola with the given characteristics. Vertex: \((0,2) ;\) directrix: \(y=4\)

Short Answer

Expert verified
The standard form of the equation of the parabola with the given characteristics is \(y=0.125x^2+2\).

Step by step solution

01

Identify vertex and directrix

The vertexes given are \(h=0\) and \(k=2\). The equation of the directrix is \(y=4\).
02

Calculate the distance between the vertex and the directrix

This can be found by taking the absolute value of the difference of the y-coordinate of the vertex and the y-coordinate of the directrix which is \(|2 - 4| = 2\).
03

Find the value of \(a\)

To find the value of \(a\) we use the formula \(a = 1/(4f)\) where \(f\) is the distance from the vertex to the directrix or the focus. Substituting the calculated distance we get \(a = 1/(4*2) = 0.125\).
04

Substitute \(h\), \(k\), and \(a\) into the standard form of parabola.

Substituting \(h=0\), \(k=2\) and \(a=0.125\) in to the parabola equation \(y=a(x-h)^2+k\) we get the equation \(y=0.125x^2+2\). This is the standard form of the equation of the parabola with the given characteristics.

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Most popular questions from this chapter

Determine whether the statement is true or false. Justify your answer. It is possible for a parabola to intersect its directrix.

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Find the distance between the point and the line. Point \((3,2)\) Line y=2 x-1

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