Question

13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.

4x² - y² = 16

Accepted Answer

Rewrite the given equation 4x² - y² = 16 in a standard form by dividing both sides by 16 to isolate the terms: $$\frac{4x$$^{2}$$}{16} - \frac{y$$^{2}$$}{16} = 1. $$Simplify the fractions to get $$\frac{x$$^{2}$$}{4} - \frac{y$$^{2}$$}{16} = 1$$, which matches the standard form of a hyperbola $$\frac{x$$^{2}$$}{a$$^{2}$$} - \frac{y$$^{2}$$}{b$$^{2}$$} = 1$$ where $$a^{2} = 4$ and b$$^{2}$$ = 16. $$Identify the center of the hyperbola at the origin (0,0), and note that since the $$x^{2}$$ term is positive, the hyperbola opens left and right along the x-axis. Find the vertices by using $$a = \sqrt{4} = 2$$, so the vertices are at $$(\pm 2, 0). $$Calculate the foci using $$c = \sqrt{a$$^{2}$$ + b$$^{2}$$} = \sqrt{4 + 16} = \sqrt{20}$$, so the foci are at $$(\pm \sqrt{20}, 0)$$, and find the equations of the asymptotes using $$y = \pm \frac{b}{a} x = \pm \frac{4}{2} x = \pm 2x.$$

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Key Concepts

Identification of Conic Sections from Equations

Standard Forms and Properties of Conic Sections

Graphing and Key Features of Hyperbolas