How to Analyze a Hyperbola

To analyze a hyperbola, we need to understand its properties and characteristics. A hyperbola is a conic section, which means it is the intersection of a cone and a plane. It has two branches that open up and down or left and right, and it is symmetrical about its center. Here are the steps to analyze a hyperbola:

1. Determine the center: The center of a hyperbola is the point where the two branches intersect. The standard form of a hyperbola is ((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1 or ((y - k)^2 / b^2) - ((x - h)^2 / a^2) = 1, where (h, k) is the center of the hyperbola.

2. Find the vertices: The vertices are the endpoints of the major axis, which is the longest axis of the hyperbola. They lie on the transverse axis, which is perpendicular to the conjugate axis. The distance between the center and the vertices is a, which is the distance from the center to the asymptotes. To find the vertices, add and subtract a from the x-coordinate or y-coordinate of the center, depending on whether the hyperbola is horizontal or vertical.

3. Find the foci: The foci are the points inside the hyperbola that are equidistant from the vertices. They lie on the transverse axis and the distance between the foci and the center is c, where c^2 = a^2 + b^2. To find the foci, add and subtract c from the x-coordinate or y-coordinate of the center, depending on whether the hyperbola is horizontal or vertical.

4. Determine the asymptotes: The asymptotes are the lines that the hyperbola approaches as the x or y values become very large or very small. They intersect at the center of the hyperbola and have slopes of ± b/a or ± a/b, depending on whether the hyperbola is horizontal or vertical. To find the equations of the asymptotes, use the center and the slopes.

5. Sketch the hyperbola: Sketch the hyperbola, including the center, vertices, foci, and asymptotes, using the information found in steps 1-4.

6. Determine the domain and range: The domain and range are the sets of all possible x and y values that the hyperbola can take. The domain is all real numbers except the x-values of the vertices, and the range is all real numbers except the y-values of the vertices.

7. Analyze the behavior of the hyperbola: The behavior of the hyperbola depends on the values of a and b. If a > b, the hyperbola is horizontal and opens left and right. If b > a, the hyperbola is vertical and opens up and down. The hyperbola approaches the asymptotes as the x or y values become very large or very small, and it approaches the x- or y-axis as the distance from the center to a point on the hyperbola approaches infinity.