Question

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. y2/4−x2/1=1

Accepted Answer

Identify the standard form of the hyperbola equation. The given equation is $$\frac{y$$^{2}$$}{4} - \frac{x$$^{2}$$}{1} = 1$$, which matches the form $$\frac{y$$^{2}$$}{a$$^{2}$$} - \frac{x$$^{2}$$}{b$$^{2}$$} = 1. $$This indicates a vertical transverse axis. Determine the values of $$a^{2}$$ and $$b^{2}. $$From the equation, $$a^{2} = 4$ and b$$^{2}$$ = 1. $$Then find $$a$$ and $$b by $$taking square roots: $$a = \sqrt{4}$$ and $$b = \sqrt{1}. $$Find the vertices of the hyperbola. Since the transverse axis is vertical, the vertices are located at $$(0, \pm a)$$, which means the vertices are at $$(0, \pm \sqrt{4}). $$Calculate the focal distance $$c$$ using the relationship $$c^{2}$$ = $$a^{2}$$ + $$b^{2}. $$Substitute the known values to find $$c^{2} = 4 + 1$ and then find c$$ = \sqrt{5}$$. Locate the foci of the hyperbola. Because the transverse axis is vertical, the foci are at (0$$, \pm c)$$, which means the foci are at (0$$, \pm \sqrt{5})$$. Use this information to match the equation to the correct graph among options (a)–(d).

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.
y²/4−x²/1=1

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주요 개념

Standard Form of a Hyperbola

Vertices of a Hyperbola

Foci of a Hyperbola