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 31
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)
Verified step by step guidance1
Identify the center of the ellipse, which is given as (0, 0). This means the ellipse is centered at the origin.
Since the major axis is horizontal with length 8, find the value of \( a \), which is half the length of the major axis: \( a = \frac{8}{2} = 4 \).
The length of the minor axis is 4, so find the value of \( b \), which is half the length of the minor axis: \( b = \frac{4}{2} = 2 \).
Because the major axis is horizontal, the standard form of the ellipse equation centered at the origin is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Substitute \( a = 4 \) and \( b = 2 \) into the equation.
Write the equation explicitly with the substituted values:
\[ \frac{x^2}{4^2} + \frac{y^2}{2^2} = 1 \]
This is the standard form of the ellipse equation with the given conditions.
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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 horizontal major axis centered at (h, k), the equation is ((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1, where '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 2a and 2b respectively, where 'a' > 'b'. Knowing these lengths allows you to find 'a' and 'b', which are essential for writing the ellipse equation.
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5:12Graph Ellipses at Origin
Center of the Ellipse
The center of the ellipse (h, k) is the midpoint of both the major and minor axes. It shifts the ellipse from the origin if not at (0, 0). In this problem, the center is at (0, 0), simplifying the equation by removing horizontal and vertical shifts.
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4:50Graph Ellipses NOT at Origin
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. Foci: (-2, 0), (2, 0); y-intercepts: -3 and 3
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
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)
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Textbook Question
Find the standard form of the equation of each hyperbola.