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Chapter 6: Problem 34

Find the vertex, focus, and directrix of the parabola. Then sketch the parabola. $$x+y^{2}=0$$

Short Answer

Expert verified
The vertex of the parabola is at (0, 0), the focus is at \(\left(-\frac{1}{4}, 0\right)\) and the equation of the directrix is \(x = \frac{1}{4}\). The parabola opens to the left.

Step by step solution

01

Find the vertex

For the given equation \(y^2 = -x\), the vertex of the parabola is at the origin (0, 0).
02

Find the value of 'a'

The standard form of the equation of a parabola is \(y^2 = 4ax\). Compare this with the given equation \(y^2 = -x\), it reveals \(4a = -1\). So, \(a = -\frac{1}{4}\).
03

Find the focus

For a parabola that opens to the left or right, the focus is \((h-a, k)\). Here, \(h = k = 0\) (from the vertex) and \(a = -\frac{1}{4}\). So, the focus is \(\left(-\frac{1}{4}, 0\right)\).
04

Find the directrix

The equation of directrix for a parabola that opens to the left or right is \(x = h + a\). Here, \(h = 0\) and \(a = -\frac{1}{4}\). The equation of the directrix is therefore \(x = \frac{1}{4}\).
05

Sketch the Parabola

Plot the vertex, focus and draw the directrix line on the x-y plane. The parabola opens towards the left since 'a' is negative, with the vertex at the origin, focus at \(\left(-\frac{1}{4}, 0\right)\) and the directrix at \(x = \frac{1}{4}\).

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