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.  x2/4−y2/1=1

Accepted Answer

Identify the standard form of the hyperbola equation given: $$\frac{x$$^{2}$$}{a$$^{2}$$} - \frac{y$$^{2}$$}{b$$^{2}$$} = 1. $$Here, $$a^{2} = 4$ and b$$^{2}$$ = 1. $$Find the vertices of the hyperbola. Since the $$x^{2}$$ term is positive and comes first, the hyperbola opens left and right. The vertices are located at $$(\pm a, 0)$$, so calculate $$a = \sqrt{4}. $$Calculate the coordinates of the vertices using the value of $$a. $$The vertices will be at $$(\pm a, 0). $$Find the foci of the hyperbola. Use the relationship $$c^{2}$$ = $$a^{2}$$ + $$b^{2} to $$find $$c$$, where $$c is $$the distance from the center to each focus. Calculate $$c = \sqrt{a$$^{2}$$ + b$$^{2}$$}$$ and then write the coordinates of the foci as $$(\pm c, 0). $$Use these points to match the hyperbola 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.
x²/4−y²/1=1

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Key Concepts

Standard Form of a Hyperbola

Vertices of a Hyperbola

Foci of a Hyperbola