In Exercises 27–32, find the standard form of the equation of each hyperbola.

Verified step by step guidance

1. Identify the center of the hyperbola, which is at the origin (0,0) based on the graph.
2. Determine the distances from the center to the vertices along the x-axis and y-axis. Here, the vertices are at (3,0) and (-3,0), so a = 3.
3. Identify the distances from the center to the co-vertices along the y-axis. Here, the co-vertices are at (0,5) and (0,-5), so b = 5.
4. Use the relationship c^2 = a^2 + b^2 to find c, where c is the distance from the center to the foci.
5. Write the standard form of the equation of the hyperbola, which is (x^2/a^2) - (y^2/b^2) = 1, and substitute the values of a and b.

Key Concepts

Standard Form of a Hyperbola

The standard form of a hyperbola centered at the origin is given by the equations (x^2/a^2) - (y^2/b^2) = 1 for horizontal hyperbolas and (y^2/a^2) - (x^2/b^2) = 1 for vertical hyperbolas. Here, 'a' represents the distance from the center to the vertices along the transverse axis, while 'b' represents the distance to the asymptotes. Understanding this form is crucial for identifying the hyperbola's orientation and dimensions.

Asymptotes of a Hyperbola

Asymptotes are lines that the hyperbola approaches but never touches. For a hyperbola centered at the origin, the equations of the asymptotes can be derived from the standard form. They are given by y = (b/a)x and y = -(b/a)x for horizontal hyperbolas, and x = (a/b)y and x = -(a/b)y for vertical hyperbolas. These lines help in sketching the hyperbola and understanding its behavior at infinity.

Vertices and Foci of a Hyperbola

The vertices of a hyperbola are the points where the hyperbola intersects its transverse axis, located at (±a, 0) for horizontal hyperbolas and (0, ±a) for vertical hyperbolas. The foci, which are points that help define the hyperbola's shape, are located at (±c, 0) for horizontal and (0, ±c) for vertical hyperbolas, where c = √(a^2 + b^2). Knowing the positions of the vertices and foci is essential for graphing the hyperbola accurately.