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Chapter 9: Problem 70

Find the equation of the parabola in standard form that satisfies the conditions given: vertex: (-3,-4) focus: (-3,-1)

Short Answer

Expert verified
The equation is \((x + 3)^2 = 12(y + 4)\).

Step by step solution

01

Understand the Vertex and Focus

The vertex of the parabola is given as \((-3, -4)\), and the focus is \((-3, -1)\). From this, we know the parabola is vertical (since the x-coordinates are the same). This means the parabola opens either up or down.
02

Calculate the Distance p

The distance \(p\) from the vertex to the focus can be calculated using the y-coordinates: \(p = -1 - (-4) = 3\). Since the focus is above the vertex, the parabola opens upwards, and \(p\) is positive.
03

Use the Standard Form Equation

The standard form of a vertical parabola is \((x - h)^2 = 4p(y - k)\), where \((h, k)\) is the vertex. Substituting \(h = -3\), \(k = -4\), and \(p = 3\), we get \((x + 3)^2 = 12(y + 4)\).
04

Expand to Verify

For verification, expand \((x + 3)^2 = 12(y + 4)\) to check its correctness: Expand to \(x^2 + 6x + 9 = 12y + 48\). Rearrange to standard form: \[ x^2 + 6x - 12y - 39 = 0 \]This confirms the parabola's equation in expanded form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertex and Focus of Parabola
The vertex and the focus are fundamental components that define a parabola. In this scenario, your problem gives you a vertex (-3, -4) and a focus (-3, -1).
- **Vertex:** This is the point on the parabola that represents its peak or trough, depending on its orientation. It can be thought of as the 'central point.'
- **Focus:** A unique point inside the parabola used to define and construct the curve. It determines how "open" or "narrow" the parabola will be.
When determining the orientation of the parabola from the vertex and focus, observe the coordinates. If the x-coordinates are identical, as they are here, the parabola is vertical. In this case, the parabola aligns with the y-axis, indicating it opens either up or down.
Parabola Opens Upwards
Understanding in which direction a parabola opens is crucial. From the details provided:
1. **Vertical Parabolas**: They can either open upwards or downwards, depending solely on the relationship between the vertex and the focus.
2. **Direction**: Given our problem's vertex and focus, note the y-coordinates. The vertex is at (-3, -4) while the focus is at (-3, -1). Since the focus is above the vertex, the parabola must open upwards.
3. **Distance p**: The distance 'p' is calculated between the vertex and the focus. For these coordinates, p = -1 - (-4) = 3. A positive value confirms the parabola opens upwards. Why? Because in parabolas, a positive 'p' value indicates the opening direction is towards the positive y-axis.
Standard Form of Parabola
The standard form for a vertical parabola illustrates its relationship with its vertex and focus. For vertical parabolas, the standard form is: \[(x-h)^2 = 4p(y-k)\] where (h, k) represents the vertex.
To apply it to the problem:
  • **Vertex values:** (h, k) = (-3, -4)
  • **Value of p:** Calculated earlier as p = 3
Substitute these into the equation: \[(x + 3)^2 = 12(y + 4)\] This gives the equation of your parabola in its standard form. To confirm the equation, expanding and rearranging results in \[x^2 + 6x - 12y - 39 = 0\]. This expresses the same equation, verifying its accuracy.

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Most popular questions from this chapter

The Witch of Agnesi: \(x(t)=2 k t ; y(t)=\frac{2 k}{1+t^{2}}\)The Witch of Agnesi is a parametric curve named by Maria Agnesi in \(1748 .\) Some believe she confused the Italian word for witch (versiera), with a similar word that meant free to move. In any case, the name stuck. The curve can also be stated in trigonometric form: \(x(t)=2 k \cot t\) and \(y=2 k \sin ^{2} t\) . a. Graph the curve with \(k=1\) on a calculator or computer on a reduced window ( \((200 \mathrm{m})\) 4) using both of the forms shown with Tmin \(=-6,\) Tmax \(=6,\) and Tstep \(=0.1 .\) Try to determine the maximum value. b. Explain why the \(x\) -axis is a horizontal asymptote. c. Experiment with different values of \(k\) and generalize its effect on the basic graph.

Planetary motion: The perihelion, aphelion, and orbital period of the planets Jupiter, Saturn, Uranus, and Neptune are shown in the table. Use the information to answer or complete the following exercises. The formula $$L=2 \pi \sqrt{0.5\left(a^{2}+b^{2}\right)}$$ can be used to estimate the length of the orbital path. Recall for an cllipse, \(c^{2}=a^{2}-b^{2}\). $$\begin{array}{|l|c|c|c|} \hline \text { Planet } & \begin{array}{c} \text { Perihelion } \\ \left(10^{6} \mathrm{mi}\right) \end{array} & \begin{array}{c} \text { Aphelion } \\ \left(10^{6} \mathrm{mi}\right) \end{array} & \begin{array}{c} \text { Period } \\ (y r) \end{array} \\ \hline \text { Jupiter } & 460 & 507 & 11.9 \\ \hline \text { Saturn } & 840 & 941 & 29.5 \\ \hline \text { Uranus } & 1703 & 1866 & 84 \\ \hline \text { Neptune } & 2762 & 2824 & 164.8 \\ \hline \end{array}$$ Find the eccentricity of the planets Jupiter and Saturn.

The Folium of Descartes: \(x(t)=\frac{3 k t}{1+t^{3}} ; y(t)=\frac{3 k t^{2}}{1+t^{3}}\) The Folium of Descartes is a parametric curve developed by Descartes in order to test the ability of Fermat to find its maximum and minimum values. a. Graph the curve on a graphing calculator with \(k=1\) using a reduced window \((\text { zoom } 4),\) with Tmin \(=-6,\) Tmax \(=6,\) and Tstep \(=0.1\) Locate the coordinates of the tip of the folium (the loop). b. This graph actually has a discontinuity (a break in the graph). At what value of \(t\) does this occur? c. Experiment with different values of \(k\) and generalize its effect on the basic graph.

Planetary motion: The perihelion, aphelion, and orbital period of the planets Jupiter, Saturn, Uranus, and Neptune are shown in the table. Use the information to answer or complete the following exercises. The formula $$L=2 \pi \sqrt{0.5\left(a^{2}+b^{2}\right)}$$ can be used to estimate the length of the orbital path. Recall for an cllipse, \(c^{2}=a^{2}-b^{2}\). $$\begin{array}{|l|c|c|c|} \hline \text { Planet } & \begin{array}{c} \text { Perihelion } \\ \left(10^{6} \mathrm{mi}\right) \end{array} & \begin{array}{c} \text { Aphelion } \\ \left(10^{6} \mathrm{mi}\right) \end{array} & \begin{array}{c} \text { Period } \\ (y r) \end{array} \\ \hline \text { Jupiter } & 460 & 507 & 11.9 \\ \hline \text { Saturn } & 840 & 941 & 29.5 \\ \hline \text { Uranus } & 1703 & 1866 & 84 \\ \hline \text { Neptune } & 2762 & 2824 & 164.8 \\ \hline \end{array}$$ Find the eccentricity of the planets Uranus and Neptune.

Find the vertex, focus, and directrix for the parabolas defined by the equations given, then use this information to sketch a complete graph (illustrate and name these features). For Exercises 43 to 60 , also include the focal chord. $$y^{2}=20 x$$
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