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Chapter 6: Problem 25

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix: \(y=1\)

Short Answer

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

Step by step solution

01

Identify the features of the parabola

Given that the directrix is \(y=1\) and the vertex is at the origin (0,0), it is known that the focus of the parabola is at \(y=-1\). This implies that the parabola opens downwards.
02

Find the value of 'a'

The value 'a' is found using the formula \(a=-\frac{1}{4f}\), where 'f' is the focal length(which is the distance from the vertex to the focus or the vertex to the directrix). Here, the focal length is 1 (the distance from the origin to either the focus at y=-1 or the directrix at y=1), therefore \(a=-\frac{1}{4(1)}=-\frac{1}{4}\).
03

Write the standard form of the equation

The standard form of the equation for a parabola with its vertex at the origin is \(y=ax^2\). Substituting the value of 'a' from Step 2, this becomes \(y=-\frac{1}{4}x^2\), which is the standard form of the equation of the parabola.

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