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.
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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 29
Find the standard form of the equation of each ellipse satisfying the given conditions. Foci: (-2, 0), (2, 0); y-intercepts: -3 and 3
Verified step by step guidance1
Identify the center of the ellipse. Since the foci are at (-2, 0) and (2, 0), the center is the midpoint of the foci, which is at (0, 0).
Determine the orientation of the ellipse. The foci lie on the x-axis, so the major axis is horizontal.
Calculate the distance between the center and each focus, denoted as \(c\). Here, \(c = 2\) because the foci are at \(\pm 2\) on the x-axis.
Use the y-intercepts to find the length of the minor axis. The y-intercepts are at \(y = \pm 3\), so the minor axis length is \(2b = 6\), giving \(b = 3\).
Apply the relationship between \(a\), \(b\), and \(c\) for ellipses: \(c^2 = a^2 - b^2\). Use this to solve for \(a^2\), then write the standard form equation of the ellipse centered at the origin with a horizontal major axis: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
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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
The standard form of an ellipse equation depends on the orientation of its major axis. For a horizontal major axis centered at the origin, it is (x²/a²) + (y²/b²) = 1, where 'a' is the semi-major axis and 'b' is the semi-minor axis. Understanding this form helps in writing the ellipse equation once 'a' and 'b' are known.
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Graph Ellipses at Origin
Foci and Relationship to Axes
The foci of an ellipse are two fixed points used to define the curve. The distance from the center to each focus is 'c', and it relates to the axes by the equation c² = a² - b² for horizontal ellipses. Knowing the foci helps determine 'c', which is essential to find 'a' and 'b'.
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5:30Foci and Vertices of an Ellipse
Intercepts and Their Role in Determining Axis Lengths
The y-intercepts of an ellipse give the points where the ellipse crosses the y-axis, indicating the length of the minor axis. Since the ellipse crosses the y-axis at ±b, the intercepts help find the value of 'b'. This information, combined with 'c', allows solving for 'a' and writing the equation.
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4:36Determinants of 2×2 Matrices
Related Practice
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Find the standard form of the equation of each parabola satisfying the given conditions. Focus: (- 3, 4); Directrix: y = 2
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Find the standard form of the equation of each hyperbola.
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Find the standard form of the equation of each hyperbola.
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