Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. $9x^2-4y^2=36$

Verified step by step guidance

1. Rewrite the given equation in standard form by dividing both sides by 36: \(\frac{9x^2}{36} - \frac{4y^2}{36} = \frac{36}{36}\), which simplifies to \(\frac{x^2}{4} - \frac{y^2}{9} = 1\).
2. Identify the values of \(a^2\) and \(b^2\) from the standard form: here, \(a^2 = 4\) and \(b^2 = 9\). Since the \(x^2\) term is positive, the hyperbola opens left and right along the x-axis.
3. Find the vertices using \(a\): vertices are located at \((\pm a, 0)\), so calculate \(a = \sqrt{4}\) and write the vertices as \((\pm 2, 0)\).
4. Calculate the foci using \(c\), where \(c^2 = a^2 + b^2\). Find \(c = \sqrt{4 + 9}\) and write the foci coordinates as \((\pm c, 0)\).
5. Write the equations of the asymptotes using the formula \(y = \pm \frac{b}{a} x\). Substitute \(a\) and \(b\) to get \(y = \pm \frac{3}{2} x\).

Key Concepts

Standard Form of a Hyperbola

A hyperbola's equation can be written in standard form as (x-h)^2/a^2 - (y-k)^2/b^2 = 1 or (y-k)^2/a^2 - (x-h)^2/b^2 = 1. This form helps identify the center (h, k), the orientation (horizontal or vertical), and the values of a and b, which are essential for graphing and finding key features like vertices and asymptotes.

Vertices and Foci of a Hyperbola

Vertices are points on the hyperbola closest to the center, located a units from the center along the transverse axis. Foci lie further out, at a distance c from the center, where c^2 = a^2 + b^2. Knowing vertices and foci helps in accurately sketching the hyperbola and understanding its shape.

Equations of Asymptotes for a Hyperbola

Asymptotes are lines that the hyperbola approaches but never touches. For a hyperbola centered at (h, k), the asymptotes have equations y - k = ±(b/a)(x - h) for horizontal transverse axis, or y - k = ±(a/b)(x - h) for vertical transverse axis. These lines guide the shape and direction of the hyperbola's branches.