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Chapter 7: Problem 9

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

Short Answer

Expert verified
The focus of the parabola is at (0,3) and the directrix is the line y = -3. Graph of the parabola opens upwards with the vertex at the origin.

Step by step solution

01

Rearrange the Equation to Standard Form

First, rewrite the equation into standard form. The given equation is \(x^2 = 12y\). We need to compare it with the standard form \(x^2 = 4ay\), and by comparing, we can find that the value of \(4a\) is 12.
02

Find the Value of 'a'

Find the value of 'a' by setting \(4a = 12\), the coefficient of y in the given equation. Solving this equation gives \(a = 3\).
03

Find the Focus and Directrix

For a parabola in the form \(x^2 = 4ay\), the focus of the parabola is at the point (0, a), so in this case, (0,3). The equation of the directrix is \(y = -a\), so in this case, \(y = -3\).
04

Graph the Parabola

Graph the equation using the function \(f(x) = x^2 /4a\). The vertex is at the origin (0,0), the focus is at (0,3) and the directrix is the line \(y = -3\). Make sure to also draw the axis of symmetry, which is the vertical line passing through the vertex and the focus.

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