Find an equation for the hyperbola described. Graph the equation. Vertices at (1,-3) and (1,1) a

step 1 Hi, in this question we have to find the equation of the hyperbola and we have to graph that equ

Find an equation for the hyperbola described. Graph the...

Key Concepts

Hyperbola

A hyperbola is a type of conic section defined as the set of all points in the plane where the absolute difference of the distances to two fixed points (foci) is constant. Hyperbolas have two separate branches and are characterized by a center, vertices, asymptotes, and axes (transverse and conjugate). Recognizing the general properties of hyperbolas is essential for identifying and graphing them as well as for formulating their equations in standard form.

Vertices

The vertices of a hyperbola are the points where the hyperbola intersects its transverse axis. They are key in determining the center of the hyperbola, which is the midpoint between the vertices, and they indicate the direction in which the hyperbola opens. Knowing the location of the vertices allows you to calculate the distance 'a', which is the distance from the center to each vertex along the transverse axis.

Center

The center of a hyperbola is the midpoint of the line segment joining the vertices. It is an essential component in the hyperbola's standard equation and serves as the point of symmetry for the hyperbola. Determining the center is the first step in translating the hyperbola to its standard form, making subsequent calculations of other parameters such as a (distance to vertices) and b (related to the asymptote slopes) straightforward.

Transverse Axis

The transverse axis of a hyperbola is the line that passes through the center and the vertices. It defines the direction in which the hyperbola opens (vertically or horizontally). The length of the transverse axis is determined by the distance between the vertices, and it provides the basis for determining the value of 'a' in the equation of the hyperbola.

Asymptotes

Asymptotes are lines that the branches of a hyperbola approach but never intersect. They provide important information about the slope and orientation of the hyperbola. In the standard form of a hyperbola's equation, the slopes of the asymptotes help in determining the value of 'b' relative to 'a'. Graphically, plotting the asymptotes gives a framework within which the hyperbola is drawn, ensuring accuracy in shape and direction.

Standard Form Equation

The standard form equation of a hyperbola expresses the relationship between x and y based on its center (h, k), the distance 'a' to the vertices, and the distance 'b' which relates to the asymptote slopes. For a vertical hyperbola, the equation typically has the form ((y-k)²/a²) - ((x-h)²/b²) = 1, while for a horizontal hyperbola it is ((x-h)²/a²) - ((y-k)²/b²) = 1. Understanding this form is crucial for both algebraic manipulation and accurate graphing of the hyperbola.

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