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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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).

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Standard Form of a Hyperbola

A hyperbola's equation in standard form is either (x²/a²) - (y²/b²) = 1 or (y²/a²) - (x²/b²) = 1. This form helps identify the orientation (horizontal or vertical) and key parameters like a and b, which are essential for locating vertices and foci.

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Asymptotes of Hyperbolas

Vertices of a Hyperbola

Vertices are points where the hyperbola intersects its transverse axis. For the equation (x²/a²) - (y²/b²) = 1, vertices are at (±a, 0). Knowing the vertices helps in sketching the hyperbola and understanding its shape and size.

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5:22

Foci and Vertices of Hyperbolas

Foci of a Hyperbola

Foci are two fixed points used to define a hyperbola, located along the transverse axis at a distance c from the center, where c² = a² + b². Finding the foci is crucial for graphing and matching the hyperbola to its correct graph.

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5:22

Foci and Vertices of Hyperbolas

Pratica correlata

Domanda del libro di testo

Find the focus and directrix of each parabola 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 = 4x

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Domanda del libro di testo

Graph the ellipse and locate the foci. $\frac{x^{2}}{36} + \frac{y^{2}}{25} = 1$