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 each ellipse and locate the foci. x2/16+y2/4 = 1
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, \(a^{2} = 16\) and \(b^{2} = 4\).
Determine which axis is the major axis by comparing \(a^{2}\) and \(b^{2}\). Since \(16 > 4\), the major axis is along the x-axis, and \(a = 4\), \(b = 2\).
Calculate the focal distance \(c\) using the relationship \(c^{2} = a^{2} - b^{2}\). Substitute the values to get \(c^{2} = 16 - 4\).
Find the value of \(c\) by taking the square root: \(c = \sqrt{12}\). This gives the distance from the center to each focus along the major axis.
Locate the foci at points \((\pm c, 0)\) on the x-axis, and sketch the ellipse centered at the origin with vertices at \((\pm a, 0)\) and co-vertices at \((0, \pm b)\).
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Standard Form of an Ellipse
An ellipse in standard form is written as (x²/a²) + (y²/b²) = 1, where a and b are the lengths of the semi-major and semi-minor axes. Understanding this form helps identify the shape and orientation of the ellipse on the coordinate plane.
Video consigliato:
5:12
Graph Ellipses at Origin
Major and Minor Axes
The major axis is the longest diameter of the ellipse, while the minor axis is the shortest. The values of a and b determine which axis is major or minor, affecting the ellipse's width and height and guiding accurate graphing.
Video consigliato:
5:12Graph Ellipses at Origin
Foci of an Ellipse
The foci are two fixed points inside the ellipse such that the sum of distances from any point on the ellipse to the foci is constant. Their positions are found using c² = a² - b², where c is the distance from the center to each focus along the major axis.
Video consigliato:
5:30Foci and Vertices of an Ellipse
Pratica correlata
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Graph the ellipse and locate the foci.
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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
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