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Magazine Chapter 8: Problem 36
For the following exercises, graph the parabola, labeling the focus and the directrix $$ -5(x+5)^{2}=4(y+5) $$
Short Answer
Expert verified
The parabola has vertex (-5, -5), opens downwards, focus (-5, -6), and directrix y = -4.
Step by step solution
01
Recognize the Parabola Form
The equation given \[-5(x+5)^{2}=4(y+5)\]is of the form \[a(x-h)^{2}=4p(y-k)\]which represents a vertical parabola centered at \((h, k)\). In this equation, \[a = -5, \quad h = -5, \quad k = -5.\]
02
Determine the Orientation and Parameter
Since the coefficient of \((x+5)^{2}\)is negative, the parabola opens downwards. Comparing with the standard form, we have \[4p = 4.\]From this, solve for \(p\):\[p = \frac{1}{1} = 1.\]
03
Identify the Vertex
The vertex \((h, k)\)is given by \((-5, -5)\).
04
Find the Focus
The focus of a parabola is located a distance \(p\) from the vertex along the axis of symmetry. Since the parabola is vertical and opens downward, compute the focus \((h, k-p)\):\((-5, -5 -1) = (-5, -6).\)
05
Determine the Directrix
The directrix of a vertical parabola is a horizontal line located at \(k+p\) from the vertex. Since \(k = -5\) and \(p=1\), the equation of the directrix is \(y = -5 + 1 = -4.\)
06
Graph the Parabola
Plot the vertex \((-5, -5)\), focus \((-5, -6)\), draw the directrix \(y = -4\), and sketch the downward-opening parabola with its axis of symmetry along the line \(x = -5\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Parabolas
Graphing a parabola can seem daunting at first, but breaking it down step by step makes it much simpler. When you're given a quadratic equation, the first step is to identify its type and orientation. In the case of \[-5(x+5)^{2}=4(y+5),\]we see this is a vertical parabola because it follows the format \[a(x-h)^{2}=4p(y-k).\] To graph this parabola:
- Start by finding the key features like the vertex, focus, and directrix.
- Determine the orientation of the parabola; here, it opens downwards due to the negative coefficient of \((x+5)^{2}\).
- Plot these points and lines on a graph to help visualize the parabola's path.
Focus and Directrix
The focus and directrix are two critical components in defining the shape and position of a parabola. They are used to derive the set of points that make up the parabola. Here's how to find them for a vertical parabola like ours:
- Focus: The focus is a unique point inside the parabola that is the same fixed distance from any parabola point as the directrix. For \[-5(x+5)^{2}=4(y+5)\], calculate the focus by moving \(p\) units vertically from the vertex. Since the parabola opens downwards, the focus is at \((-5, -6)\).
- Directrix: This is a horizontal line located at a distance \(p\) from the vertex in the opposite direction of the parabola's opening. For this equation, the directrix is \(y = -4\).
Vertex of a Parabola
The vertex is perhaps the most vital part of a parabola as it indicates the maximum or minimum point, from which the parabola curves away. In the equation\[-5(x+5)^{2}=4(y+5),\]the vertex is found at \((h, k) = (-5, -5).\) This spot is crucial as it serves as the reference point when determining the focus and directrix.
Always remember:
Always remember:
- The vertex is the middle point of the parabola's curve.
- The axis of symmetry goes through the vertex; for vertical parabolas, it's a vertical line \(x = h.\)
- It helps to divide the parabola into symmetrical halves.
Orientation of Parabolas
The orientation of the parabola tells you the direction in which it opens. For the equation \[-5(x+5)^{2}=4(y+5),\]we know the parabola opens downwards due to the negative coefficient \(-5\) appearing before the squared term. Understanding the orientation is essential because:
- It affects the direction of opening; positive leads upwards, negative leads downwards.
- Helps in correctly positioning the focus and directrix.
- Affects how you draw points and the overall parabola curve.
