Domanda del libro di testo
Find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. (y + 1)2 = - 8x
764
views
Ch. 7 - Conic Sections
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
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 41
Graph each ellipse and give the location of its foci. (x − 4)²/9 + (y +2)² /25= 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 \(\frac{(x - 3)^2}{4} + \frac{(y + 1)^2}{16} = 1\), determine the center as \((3, -1)\).
Compare the denominators to find \(a^2\) and \(b^2\). Here, \(a^2 = 16\) and \(b^2 = 4\). Since \(a^2 > b^2\), the major axis is vertical.
Calculate the distance \(c\) from the center to each focus using the formula \(c = \sqrt{a^2 - b^2}\).
Locate the foci at \((h, k \pm c)\) because the major axis is vertical, so the foci lie above and below the center along the y-axis.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
3mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
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. Understanding this form helps in identifying the ellipse's size, shape, and position 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 larger denominator (a² or b²) corresponds to the major axis. Knowing which axis is major or minor is essential for graphing the ellipse accurately and determining the orientation (horizontal or vertical).
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 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 crucial for understanding the ellipse's geometric properties.
Video consigliato:
5:30Foci and Vertices of an Ellipse
Pratica correlata
Domanda del libro di testo
Convert each equation to standard form by completing the square on x or y. Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola. x2 - 2x - 4y + 9 =0
849
views
Domanda del libro di testo
Find the standard form of the equation of the parabola satisfying the given conditions. Focus: (12,0); Directrix: x=-12
1485
views
Domanda del libro di testo
Find the standard form of the equation of the parabola satisfying the given conditions. Focus: (0,-11); Directrix: y=11
1714
views
Domanda del libro di testo
Use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. (x−1)2−(y−2)2=3
835
views
Domanda del libro di testo
Graph each ellipse and give the location of its foci. x²/25 + (y -2)² /36= 1
837
views