Domanda del libro di testo
Find the focus and directrix of the parabola with the given equation. Then graph the parabola. 8x2 + 4y = 0
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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 15
Graph each ellipse and locate the foci.4x²+16y² = 64
Guida verificata passo dopo passo1
Rewrite the given ellipse equation \(4x^2 + 16y^2 = 64\) in standard form by dividing every term by 64 to isolate 1 on the right side: \(\frac{4x^2}{64} + \frac{16y^2}{64} = 1\).
Simplify the fractions to get the equation in the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Identify \(a^2\) and \(b^2\) from the denominators.
Determine which denominator is larger to identify the major axis. If \(a^2 > b^2\), the major axis is along the x-axis; if \(b^2 > a^2\), it is along the y-axis.
Calculate the coordinates of the vertices using \(\pm a\) or \(\pm b\) depending on the major axis. These points lie on the ellipse along the major axis.
Find the foci by calculating \(c = \sqrt{|a^2 - b^2|}\). The foci are located at \((\pm c, 0)\) if the major axis is horizontal, or \((0, \pm c)\) if vertical.
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Standard Form of an Ellipse
An ellipse equation can be written in the form (x²/a²) + (y²/b²) = 1, where a and b are the lengths of the semi-major and semi-minor axes. Converting the given equation into this form helps identify these axes and understand the ellipse's shape and orientation.
Video consigliato:
5:12
Graph Ellipses at Origin
Graphing an Ellipse
Graphing an ellipse involves plotting its center, major and minor axes based on a and b values, and sketching the curve accordingly. Knowing the lengths and orientation of these axes allows accurate representation of the ellipse on the coordinate plane.
Video consigliato:
4:50Graph Ellipses NOT 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
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