Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. x²/9−y²/25=1

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Identify the standard form of the hyperbola equation given: \(\frac{x^{2}}{9} - \frac{y^{2}}{25} = 1\). This is a hyperbola centered at the origin with the \(x^{2}\) term positive, indicating it opens left and right.

Determine the values of \(a^{2}\) and \(b^{2}\) from the denominators: \(a^{2} = 9\) and \(b^{2} = 25\). Then find \(a = \sqrt{9} = 3\) and \(b = \sqrt{25} = 5\).

Locate the vertices on the \(x\)-axis at \((\pm a, 0)\), which are \((\pm 3, 0)\), since the hyperbola opens horizontally.

Find the foci using the relationship \(c^{2} = a^{2} + b^{2}\). Calculate \(c = \sqrt{9 + 25}\), then place the foci at \((\pm c, 0)\) along the \(x\)-axis.

Write the equations of the asymptotes for a hyperbola centered at the origin with horizontal transverse axis: \(y = \pm \frac{b}{a} x\). Substitute \(a\) and \(b\) to get \(y = \pm \frac{5}{3} x\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 of the hyperbola (horizontal or vertical), the lengths of the transverse and conjugate axes, and the locations of vertices and asymptotes.

Vertices and Foci of a Hyperbola

Vertices are points where the hyperbola intersects its transverse axis, located at a distance 'a' from the center. Foci lie further along this axis at a distance 'c', where c² = a² + b². Knowing vertices and foci is essential for graphing and understanding the hyperbola's shape.

Equations of Asymptotes for a Hyperbola

Asymptotes are lines that the hyperbola approaches but never touches. For a hyperbola centered at the origin, the asymptotes have equations y = ±(b/a)x for horizontal hyperbolas and y = ±(a/b)x for vertical ones. These lines guide the graph's shape and help in sketching the hyperbola.

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