Graph the hyperbola. Locate the foci and find the equations of the asymptotes. 4y^2 - x^2 = 16

Verified step by step guidance

1. Rewrite the given equation in standard form for a hyperbola. Divide through by 16 to normalize the equation: \( \frac{4y^2}{16} - \frac{x^2}{16} = 1 \), which simplifies to \( \frac{y^2}{4} - \frac{x^2}{16} = 1 \).
2. Identify the type of hyperbola. Since the \( y^2 \) term is positive and comes first, this is a vertical hyperbola. The standard form for a vertical hyperbola is \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \). Here, \( a^2 = 4 \) and \( b^2 = 16 \), so \( a = 2 \) and \( b = 4 \).
3. Determine the foci. The distance from the center to each focus is given by \( c = \sqrt{a^2 + b^2} \). Substitute \( a^2 = 4 \) and \( b^2 = 16 \) into the formula to find \( c \). The foci are located at \( (0, \pm c) \) since the hyperbola is vertical.
4. Find the equations of the asymptotes. For a vertical hyperbola, the asymptotes are given by \( y = \pm \frac{a}{b}x \). Substitute \( a = 2 \) and \( b = 4 \) into the formula to write the equations of the asymptotes.
5. Sketch the graph. Plot the center at \( (0, 0) \), draw the vertices at \( (0, \pm a) \), and sketch the asymptotes as diagonal lines passing through the center. Use the foci and the asymptotes to guide the shape of the hyperbola.

Key Concepts

Hyperbola Definition

A hyperbola is a type of conic section formed by the intersection of a plane and a double cone. It consists of two separate curves called branches, which are mirror images of each other. The standard form of a hyperbola can be expressed as (y^2/a^2) - (x^2/b^2) = 1 or (x^2/a^2) - (y^2/b^2) = 1, depending on its orientation.

Foci of a Hyperbola

The foci of a hyperbola are two fixed points located along the transverse axis, which is the line segment that connects the vertices of the hyperbola. The distance from the center to each focus is denoted as 'c', where c = √(a^2 + b^2). The foci play a crucial role in defining the shape and properties of the hyperbola, as they are used to determine the distances that define the hyperbola's branches.

Asymptotes of a Hyperbola

Asymptotes are lines that the branches of a hyperbola approach but never touch. For a hyperbola in the form (y^2/a^2) - (x^2/b^2) = 1, the equations of the asymptotes are given by y = ±(a/b)x. These lines provide a framework for sketching the hyperbola and indicate the direction in which the branches extend infinitely.