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Chapter 6: Problem 26

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix: \(y=-2\)

Short Answer

Expert verified
The standard form of the equation of the parabola with the vertex at origin and a directrix at \(y=-2\) is \(y = 1/4x^2\)

Step by step solution

01

Identify the form of the parabola

Given the information, we know that the parabola opens upwards because the directrix \(y = -2\) is a horizontal line below the vertex (0,0). Therefore, the parabola's vertex form will be \(y = ax^2\)
02

Find the distance from the vertex to the directrix

The vertex is at the origin (0,0) and the directrix is at \(y=-2\). Therefore, the distance between the vertex and the directrix is 2.
03

Calculate the value of 'a'

The value of 'a' is the inverse of 4 times the distance between the vertex and the directrix. So in this case, \(a = 1/(4*2) = 0.25\) or \(1/4\)
04

Write the equation in standard form

Substitute the value of 'a' into the standard form equation. So, the standard form equation of the parabola is \(y = 0.25x^2\) or \(y = 1/4x^2\)

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