Domanda del libro di testo
Find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. (x + 1)2 = - 8(y + 1)
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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 37
Graph each ellipse and give the location of its foci. (x − 2)²/9 + (y -1)² /4= 1
Guida verificata passo dopo passo1
Identify the standard form of the ellipse equation: \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\), where \((h, k)\) is the center of the ellipse.
From the given equation \(\frac{(x - 3)^2}{36} + \frac{(y - 4)^2}{25} = 1\), determine the center as \((3, 4)\).
Compare the denominators to find \(a^2\) and \(b^2\). Since 36 > 25, \(a^2 = 36\) and \(b^2 = 25\). This means the major axis is horizontal because \(a^2\) is under the \((x - h)^2\) term.
Calculate the lengths of the semi-major axis \(a = \sqrt{36}\) and the semi-minor axis \(b = \sqrt{25}\).
Find the distance \(c\) from the center to each focus using the formula \(c = \sqrt{a^2 - b^2}\). Then, locate the foci at \((h \pm c, k)\) because the major axis is horizontal.
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Standard Form of an Ellipse
The standard form of an ellipse equation is (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center. The values a² and b² represent the squares of the lengths of the semi-major and semi-minor axes, respectively. Understanding this form helps in identifying the ellipse's size, shape, and position on the coordinate plane.
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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 lengths of these axes are 2a and 2b, where a and b are the square roots of the denominators in the ellipse equation. Knowing which axis is major or minor depends on whether a² > b² or vice versa.
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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 locations are found using c² = a² - b², where c is the distance from the center to each focus along the major axis. Identifying the foci is essential for graphing and understanding ellipse properties.
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5:30Foci and Vertices of an Ellipse
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