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Chapter 7: Problem 37

Find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. $$(x+1)^{2}=-8(y+1)$$

Short Answer

Expert verified
The vertex is at (-1, -1), the focus is at (-1, -3), and the equation for the directrix is y = 1.

Step by step solution

01

Identify the Constants

The given equation is \((x+1)^{2} = -8(y+1)\). Comparing this with \((x-h)^{2} = 4a(y-k)\), we get 'h' as -1, 'k' as -1, and 'a' as '-2'. Hence, the parabola opens downward.
02

Find the Vertex

The vertex of the parabola will be at (h, k). Substituting the values of h and k that we identified, the vertex will be at (-1, -1).
03

Find the Focus

The focus of the parabola will be at (h, k+a). Substituting the values of h, k and a that were identified, the focus will be at (-1, -1-2) = (-1, -3).
04

Find the Directrix

The equation of the directrix will be y = k - a. Substituting the values of k and a that we identified, the equation of the directrix will be y = -1-(-2) = y = 1.
05

Graph the Parabola

Finally, using the vertex, focus, and directrix, we will be able to sketch the graph of the parabola. Remember that the parabola is a mirror image about its directrix, and is symmetrical around the vertex.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. What happens to the shape of the graph of \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) as \(\frac{c}{a} \rightarrow \infty,\) where \(c^{2}=a^{2}+b^{2} ?\)

Find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. $$(y-1)^{2}=-8 x$$

Describe one similarity and one difference between the graphs of \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\) and \(\frac{(x-3)^{2}}{9}-\frac{(y+3)^{2}}{1}=1\)

Will help you prepare for the material covered in the first section of the next chapter. Evaluate \(i^{2}+1\) for all consecutive integers from 1 to 6 inclusive. Then find the sum of the six evaluations.

Describe one similarity and one difference between the graphs of \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\) and \(\frac{(x-3)^{2}}{9}-\frac{(y+3)^{2}}{1}=1\)
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