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Chapter 7: Problem 13

Find the focus and directrix of the parabola with the given equation. Then graph the parabola. $$y^{2}-6 x=0$$

Short Answer

Expert verified
The focus of the given parabola is at point (1.5, 0) and the equation of the directrix is x = -1.5. The graph of the parabola opens to the right, passing through the focus and is parallel to the directrix.

Step by step solution

01

Rewrite Equation in Standard Form

Reframing the given equation \(y^2 - 6x = 0 \) in standard form to obtain: \(y^2=6x\)
02

Identify the form and value of 4p

Identify if the equation is in the form \(y^2 = 4px\) or \(x^2 = 4py\). Here, equation is in the form \(y^2 = 4px \) where \(4p = 6\) and hence, \(p = 1.5\). The value of p helps us in calculating the focus and directrix.
03

Determine focus and directrix

For the parabola in the form \(y^2 = 4px\), the focus is at (p,0) and directrix is at x = -p. So, for the given equation, the focus is at (1.5, 0) and directrix is at x = -1.5.
04

Graph the parabola

Plot the focus point on the x-axis and draw the directrix line at x = -1.5. Lastly, plot the parabola which will open towards the right (as the equation is in the form \(y^2 = 4px\)) and passing through the focus and parallel to the directrix.

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Most popular questions from this chapter

Describe how to locate the foci for \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\)

Wre a graphing utility to graph \(\frac{x^{2}}{4}-\frac{y^{2}}{9}=0 .\) Is the graph a hyperbola? In general, what is the graph of \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=0 ?\)

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Find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. $$(x+2)^{2}=-8(y+2)$$

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