Domanda del libro di testo
Find the 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).
a. b. c. d.
y2 = 4x
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Ch. 7 - Conic Sections
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 8, Problema 1
Graph the ellipse and locate the foci.
Guida verificata passo dopo passo1
Identify the standard form of the ellipse equation: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Here, \(a^2 = 36\) and \(b^2 = 25\).
Determine the values of \(a\) and \(b\) by taking the square roots: \(a = \sqrt{36} = 6\) and \(b = \sqrt{25} = 5\).
Since \(a > b\), the major axis is along the x-axis. The ellipse is centered at the origin \((0,0)\) with vertices at \((\pm a, 0)\), which are \((\pm 6, 0)\).
Calculate the focal distance \(c\) using the relationship \(c^2 = a^2 - b^2\). Substitute the values to find \(c^2 = 36 - 25\).
Locate the foci at \((\pm c, 0)\) along the x-axis. These points are inside the ellipse between the center and the vertices.
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Standard Form of an Ellipse
An ellipse in standard form is written as (x^2/a^2) + (y^2/b^2) = 1, where a and b are the lengths of the semi-major and semi-minor axes. Identifying a and b helps determine the shape and orientation of the ellipse on the coordinate plane.
Video consigliato:
5:12
Graph Ellipses at Origin
Graphing an Ellipse
To graph an ellipse, plot the center at the origin, then mark points a units along the major axis and b units along the minor axis. Connecting these points smoothly forms the ellipse, showing its size and orientation.
Video consigliato:
4:50Graph Ellipses NOT at Origin
Locating the Foci of an Ellipse
The foci are two fixed points inside the ellipse located along the major axis. Their distance from the center is c, found using c^2 = a^2 - b^2. Knowing c allows you to place the foci accurately on the graph.
Video consigliato:
5:30Foci and Vertices of an Ellipse
Pratica correlata
Domanda del libro di testo
In Exercises 1–4, find the 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^2 = 4y
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Domanda del libro di testo
Graph each ellipse and locate the foci. x2/16+y2/4 = 1
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Domanda del libro di testo
Find the vertices and locate the foci of each hyperbola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)–(d).
a. b. c. d.
x2/4−y2/1=1
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Domanda del libro di testo
Graph the ellipse and locate the foci. (y^2)/25 + (x^2)/16 = 1
1098
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Domanda del libro di testo
Find the vertices and locate the foci of each hyperbola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)–(d).
a. b. c. d.
y2/4−x2/1=1
875
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