Find the standard form of the equation of each hyperbola.

Verified step by step guidance

1. Identify the center of the hyperbola. From the graph, the hyperbola is centered at the origin (0, 0).
2. Determine the coordinates of the vertices. The vertices are given as (0, 7) and (0, -7), so the distance from the center to each vertex is 7. This means \(a = 7\).
3. Since the vertices lie on the y-axis, the hyperbola opens vertically. The standard form for a vertical hyperbola centered at the origin is: \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\)
4. Find the slopes of the asymptotes from the graph. The asymptotes are the dashed lines crossing through the origin. The slopes appear to be \(\pm \frac{b}{a}\). From the graph, the asymptotes pass through points (7, 5) and (5, 7), so calculate the slope as \(\frac{5}{7}\) or \(\frac{7}{5}\) and identify which corresponds to \(\frac{b}{a}\).
5. Use the slope of the asymptotes to solve for \(b\). Since \(a = 7\), and slope \(= \frac{b}{a}\), multiply the slope by \(a\) to find \(b\). Then write the standard form equation by substituting \(a^2\) and \(b^2\) into the formula.

Key Concepts

Standard Form of a Hyperbola

The standard form of a hyperbola equation depends on its orientation. For a hyperbola centered at the origin with vertical transverse axis, the form is (y²/a²) - (x²/b²) = 1. Here, 'a' is the distance from the center to the vertices along the transverse axis, and 'b' relates to the conjugate axis. Identifying 'a' and 'b' from the graph is essential to write the equation.

Vertices and Center of the Hyperbola

The vertices are points where the hyperbola intersects its transverse axis, indicating the distance 'a' from the center. The center is the midpoint between the vertices, often at the origin in this problem. Knowing the coordinates of the vertices (0,7) and (0,-7) helps determine 'a' and confirms the hyperbola's vertical orientation.

Asymptotes of a Hyperbola

Asymptotes are lines that the hyperbola approaches but never touches, defined by y = ±(a/b)x for vertical hyperbolas. The slopes of the asymptotes help find 'b' once 'a' is known. The dashed lines in the graph represent these asymptotes, and their slopes can be calculated from the grid to complete the hyperbola's equation.