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Chapter 6: Problem 41

Find the vertex, focus, and directrix of the parabola. Then sketch the parabola. $$y^{2}+6 y+8 x+25=0$$

Short Answer

Expert verified
The vertex of the parabola is at (-4,3), the focus is at (-2,3), and the directrix is the line \(x=-2\). The parabola opens to the left.

Step by step solution

01

Rearrange the given equation in the standard form

Let's try to change the given quadratic equation into the standard form (similar to \((y-k)^2 = 4p(x-h)\)).1. Rewrite the equation: \(y^2-6 y = -8x-25\)2. Next step, completing the square: \( (y-3)^2 = -8x-25+9\)3. We get the final form: \((y-3)^2 = -8(x+4)\)
02

Identify the vertex, focus, and directrix

From the standard form of the equation, vertex (h,k) can be read directly as (-4,3). The constant term with x (which is -8, or 4p) allows to find the value of p, which is -2. Remember that for a vertically-aligned parabola, p is the horizontal distance from the vertex to the focus and directrix. So, the focus of the parabola is at (h-p , k) or (-4 -(-2), 3) which becomes (-2, 3). The equation of the directrix, which is always on the opposite side of the parabola from the focus, is given by \(x=h+p\). So, the directrix is at \(x=-4-(-2)\) or \(x=-2\).
03

Sketch the Parabola

Now, plot the vertex (-4,3), focus (-2,3) and draw the vertical line \(x=-2\) for the directrix. Ensure that the parabola is opening to the left because the x-term in the standard form of the equation is negative. Connect these points and sketch the curve of the parabola around them.

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Most popular questions from this chapter

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