Find the standard form of the equation of each hyperbola.

Verified step by step guidance

1. Identify the center of the hyperbola. Since the hyperbola is centered at the origin, the center is at (0, 0).
2. Determine the orientation of the hyperbola. The branches open left and right, so it is a horizontal hyperbola.
3. Find the values of 'a' and 'b' by examining the rectangle formed by the dashed lines. The rectangle extends from -3 to 3 on the x-axis and from -5 to 5 on the y-axis, so \(a = 3\) and \(b = 5\).
4. Write the standard form of the equation for a horizontal hyperbola centered at the origin: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).
5. Substitute the values of \(a\) and \(b\) into the equation: \(\frac{x^2}{3^2} - \frac{y^2}{5^2} = 1\).

Key Concepts

Standard Form of a Hyperbola

The standard form of a hyperbola centered at the origin is either (x^2/a^2) - (y^2/b^2) = 1 for a horizontal transverse axis or (y^2/a^2) - (x^2/b^2) = 1 for a vertical transverse axis. Here, 'a' and 'b' represent distances related to the vertices and asymptotes, respectively.

Asymptotes of a Hyperbola

Asymptotes are lines that the hyperbola approaches but never touches. For hyperbolas centered at the origin, the asymptotes have equations y = ±(b/a)x for horizontal transverse axis or y = ±(a/b)x for vertical transverse axis. They help determine the shape and orientation of the hyperbola.

Graph Interpretation and Parameter Identification

Analyzing the graph involves identifying key points such as vertices and the rectangle formed by 'a' and 'b' values. The vertices indicate 'a', while the slopes of the asymptotes help find 'b'. This information is essential to write the hyperbola's equation in standard form.