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Chapter 6: Problem 32

Find the vertex, focus, and directrix of the parabola. Then sketch the parabola. $$y^{2}=3 x$$

Short Answer

Expert verified
The vertex is at (0,0), the focus is at (\(\frac{3}{4}\),0), the directrix is \(x = -\frac{3}{4}\), and the parabola opens to the right.

Step by step solution

01

Express the given equation in parabolic form.

The equation of the parabola is given as \(y^{2}=3 x\). We'll rewrite this as \(y^{2}= 4px\) and compare the two equations to find the value of \(p\).
02

Find the value of p

We compare our equation to the form \(y^{2}=4px\). By comparison, \(4p=3\), so \(p= \frac{3}{4}\). This tells us the parabola opens to the right because p is positive.
03

Identify the vertex

For this form of parabola, the vertex is always at the origin. So the vertex of the parabola is at (0,0).
04

Find the focus

In this parabola, the focus is at \((p,0)\). Since \(p=\frac{3}{4}\), the focus of the parabola is at \((\frac{3}{4},0)\).
05

Identify the directrix

The directrix in this form of a parabola is \(x = -p\). Thus, since \(p=\frac{3}{4}\), the equation of the directrix is \(x = -\frac{3}{4}\).
06

Draw the sketch of the parabola

First, plot the vertex, focus and a vertical line for the directrix. The parabola opens towards the focus, meaning it opens to the right. Then sketch the parabola such that it envelops the focus and is tangent to the directrix.

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