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Chapter 7: Problem 8

Find the focus and directrix of the parabola with the given equation. Then graph the parabola. $$y^{2}=-12 x$$

Short Answer

Expert verified
The focus of the given parabola is at the point \(-3, 0\) and the equation of directrix is \(x = 3\).

Step by step solution

01

Convert the equation into standard form

To convert the given equation into standard form, it has to be written as \(4p(x-h)= (y-k)^{2}\) for horizontal parabolas. We have \(y^{2}= -12x\), so it will be written as \(-4(3x)= y^{2}\). Comparing it to the standard form, we can say that \(h=0, k=0, p=-3\).
02

Finding the focus

For horizontal parabolas, the focus is calculated using the formula \(h+p, k\), substituting the values obtained from our previous step, our focus will be \(-3,0\).
03

Finding Directrix

For horizontal parabolas, the directrix is calculated using the formula \(x=h-p\). Substituting the obtained values, we get our directrix as \(x=3\).
04

Graph the Parabola

To graph the parabola, we need to firstly plot the vertex which is the point (h,k) obtained in step 1. Additionally, we plot the focus found in step 2. Also, draw a vertical line corresponding to the directrix equation obtained in step 3. Now, draw a smooth, continuous curve representing the parabola that touches the vertex, outlines the focus inside and is tangent to the directrix.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. In a whispering gallery at our science museum, I stood at one focus, my friend stood at the other focus, and we had a clear conversation, very little of which was heard by the 25 museum visitors standing between us.

Determine whether each statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. Some parabolas that open to the right have equations that define \(y\) as a function of \(x .\)

Convert each equation to standard form by completing the square on \(x\) or \(y .\) Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola. $$x^{2}-2 x-4 y+9=0$$

Use a graphing utility to graph the parabolas in Exercises 77-78. Write the given equation as a quadratic equation in \(y\) and use the quadratic formula to solve for \(y .\) Enter each of the equations to produce the complete graph. $$y^{2}+10 y-x+25=0$$

Convert each equation to standard form by completing the square on \(x\) or \(y .\) Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola. $$y^{2}-2 y+12 x-35=0$$
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