Chapter 9: Problem 8
Find the equation of the parabola defined by the given information. Sketch the parabola. Focus: (-1,-4)\(;\) directrix: \(y=2\)
Short Answer
Expert verified
The equation is \((x + 1)^2 = -12(y + 1)\). The parabola opens downward with vertex (-1, -1).
Step by step solution
01
Understand the Properties of the Parabola
A parabola is defined as the set of all points that are equidistant from a point called the focus and a line called the directrix. For the given problem, the focus is \((-1, -4)\) and the directrix is \(y = 2\).
02
Find the Vertex
The vertex of the parabola lies midway between the focus and the directrix. Since the focus is at \((-1, -4)\) and the directrix is \(y = 2\), the vertex \((x, y)\) can be calculated as the midpoint of the vertical distance between the focus and directrix. This results in \((-1, -1)\) because \((-4 + 2)/2 = -1\).
03
Determine the Orientation
Since the directrix is horizontal \((y = 2)\), the parabola opens either upwards or downwards. However, since the focus is below the directrix, the parabola will open downwards.
04
Find the Equation of the Parabola
The standard form of a parabola that opens vertically is \((x - h)^2 = 4p(y - k)\). Here, \((h, k)\) is the vertex of the parabola. We know \(h = -1\) and \(k = -1\). The distance \(p\) is the distance from the vertex to the focus: \(p = -3\) because the focus is 3 units below the vertex, thus the distance is negative. Substituting these into the equation gives us \((x + 1)^2 = -12(y + 1)\).
05
Sketch the Parabola
Plot the focus at \((-1, -4)\) and the horizontal line directrix at \(y = 2\). The vertex is at \((-1, -1)\) which is the midpoint vertically. Draw a downward-opening parabola with these details in mind.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Focus of a Parabola
The focus of a parabola is a fundamental point used to define the curve. This point, linked closely to how the parabola is shaped, is crucial because a parabola consists of all points that maintain an equal distance to both the focus and the directrix.
In our exercise, the focus is given as the point \((-1, -4)\). This means that each point on the parabola is equidistant from this point and a specific line, which is called the directrix. Understanding the focus helps in determining how the parabola "opens." For example, if the parabola opens downwards, the focus is below the directrix as in this instance.
Keep in mind that the placement of the focus relative to the directrix provides insight into the orientation and direction of the opening of the parabola.
In our exercise, the focus is given as the point \((-1, -4)\). This means that each point on the parabola is equidistant from this point and a specific line, which is called the directrix. Understanding the focus helps in determining how the parabola "opens." For example, if the parabola opens downwards, the focus is below the directrix as in this instance.
Keep in mind that the placement of the focus relative to the directrix provides insight into the orientation and direction of the opening of the parabola.
Vertex of a Parabola
The vertex of a parabola is a key point that represents its peak or lowest point, depending on its orientation. It is located exactly midway between the focus and the directrix, lying on the axis of symmetry of the parabola.
For example, in the current problem, the vertex is found by computing the midpoint between the focus, \((-1, -4)\), and the directrix \(y = 2\). The vertical positioning gives us the vertex at \((-1, -1)\).
The vertex is not only crucial for sketching the parabola but also serves as the reference point in the equation of the parabola. In standard form, the vertex \( h, k \) plays a vital role in shaping the equation: \((x - h)^2 = 4p(y - k)\).
For example, in the current problem, the vertex is found by computing the midpoint between the focus, \((-1, -4)\), and the directrix \(y = 2\). The vertical positioning gives us the vertex at \((-1, -1)\).
The vertex is not only crucial for sketching the parabola but also serves as the reference point in the equation of the parabola. In standard form, the vertex \( h, k \) plays a vital role in shaping the equation: \((x - h)^2 = 4p(y - k)\).
Directrix of a Parabola
The directrix is a fixed line that works together with the focus to define the shape and orientation of a parabola. The directrix guides the "stretch" and positioning of the parabola, ensuring that the distance from any point on the parabola to this line is equal to its distance to the focus.
In our scenario, the directrix is the line \(y = 2\). This is a horizontal line, meaning the parabola will open vertically. Because the focus is located below this directrix, at \((-1, -4)\), the parabola will open downward. This configuration pinpoints the vertex at an equal distance from both the directrix and the focus.
The distance between the vertex and the directrix is just as important as the distance from the vertex to the focus, helping form the parabola’s specific equation and shape.
In our scenario, the directrix is the line \(y = 2\). This is a horizontal line, meaning the parabola will open vertically. Because the focus is located below this directrix, at \((-1, -4)\), the parabola will open downward. This configuration pinpoints the vertex at an equal distance from both the directrix and the focus.
The distance between the vertex and the directrix is just as important as the distance from the vertex to the focus, helping form the parabola’s specific equation and shape.
Equation of a Parabola
The equation of a parabola reveals significant details about its orientation and the positioning of its vertex and focus. Typically, the equation is expressed in standard form as \((x - h)^2 = 4p(y - k)\) for a parabola that opens vertically.
In this exercise, the vertex \((-1, -1)\) becomes the center of this equation, plugging in as \(h = -1\) and \(k = -1\). The crucial term \(4p\) refers to the parabola's "width" parameter. The value of \(p\) denotes the distance from the vertex to the focus. In this case, since the parabola opens downwards, \(p = -3\) as the focus is 3 units lower than the vertex. Therefore, the expression transforms into \((x + 1)^2 = -12(y + 1)\).
Through this equation, one can analyze and graph the parabola accurately. It offers insights not just into the parabola's curve but also into its spatial dynamics in relation to the focus and directrix.
In this exercise, the vertex \((-1, -1)\) becomes the center of this equation, plugging in as \(h = -1\) and \(k = -1\). The crucial term \(4p\) refers to the parabola's "width" parameter. The value of \(p\) denotes the distance from the vertex to the focus. In this case, since the parabola opens downwards, \(p = -3\) as the focus is 3 units lower than the vertex. Therefore, the expression transforms into \((x + 1)^2 = -12(y + 1)\).
Through this equation, one can analyze and graph the parabola accurately. It offers insights not just into the parabola's curve but also into its spatial dynamics in relation to the focus and directrix.
