Textbook Question
Find the vertex, focus, and directrix of each parabola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)–(d). (x + 1)2 = - 4(y + 1)
a. b. c. d.
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Ch. 7 - Conic Sections
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 8, Problem 33
Find the standard form of the equation of each ellipse satisfying the given conditions. Major axis vertical with length 10; length of minor axis = 4; center: (-2, 3)
Verified step by step guidance1
Identify the center of the ellipse as \( (h, k) = (-2, 3) \).
Since the major axis is vertical, the standard form of the ellipse equation is \( \frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1 \), where \(a\) is the semi-major axis and \(b\) is the semi-minor axis.
Calculate the semi-major axis length \(a\) by dividing the major axis length by 2: \( a = \frac{10}{2} = 5 \).
Calculate the semi-minor axis length \(b\) by dividing the minor axis length by 2: \( b = \frac{4}{2} = 2 \).
Substitute \(h\), \(k\), \(a\), and \(b\) into the standard form equation: \( \frac{(x + 2)^2}{2^2} + \frac{(y - 3)^2}{5^2} = 1 \). This is the standard form of the ellipse equation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Form of an Ellipse Equation
The standard form of an ellipse equation depends on the orientation of its major axis. For a vertical major axis, the equation is ((x - h)^2 / b^2) + ((y - k)^2 / a^2) = 1, where (h, k) is the center, a is the semi-major axis length, and b is the semi-minor axis length.
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Graph Ellipses at Origin
Major and Minor Axes Lengths
The major axis is the longest diameter of the ellipse, and the minor axis is the shortest. Their lengths are twice the semi-major (a) and semi-minor (b) axes, respectively. Knowing these lengths helps determine a and b values used in the ellipse equation.
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5:12Graph Ellipses at Origin
Center of the Ellipse
The center (h, k) of the ellipse is the midpoint of both the major and minor axes. It shifts the ellipse from the origin to the point (h, k) in the coordinate plane, affecting the equation by translating x and y coordinates accordingly.
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Related Practice
Textbook Question
Find the vertex, focus, and directrix of each parabola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)–(d). (y - 1)2 = 4(x - 1)
a. b. c. d.
Textbook Question
Find the standard form of the equation of each ellipse satisfying the given conditions. Major axis horizontal with length 8; length of minor axis = 4; center: (0, 0)
Textbook Question
Use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. (x+4)2/9−(y+3)2/16=1
Textbook Question
In Exercises 31–34, find the vertex, focus, and directrix of each parabola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)–(d). (y - 1)2 = - 4(x - 1)
Textbook Question
Find the standard form of the equation of each hyperbola.