In Problems 41-48, find an equation for the hyperbola described. Graph the equation. Vertices at

step 1 So we have the equation here, so we have vertices 1, negative 3 and 1 1, and we have y plus 1, e

In Problems 41-48, find an equation for the hyperbola...

Key Concepts

Asymptotes

Asymptotes are the lines that a hyperbola approaches infinitely but never actually reaches. They reflect the hyperbola’s direction and overall spread. In the standard form, the slopes of these lines are determined by the ratio of 'a' and 'b'—specifically, they are ±(b/a) for a hyperbola with a vertical transverse axis and ±(a/b) for one with a horizontal transverse axis.

Relationship_Between_a_and_b

The constants 'a' and 'b' define the shape of the hyperbola: 'a' is the distance from the center to each vertex, and 'b' relates to the slope of the asymptotes. When one of these values is known, the other can be determined using the slope of the asymptotes given by the ratio b/a (or a/b, depending on the orientation). This relationship is fundamental to fully defining the hyperbola's equation.

Vertices

Vertices are the points where the hyperbola intersects its transverse axis. They are a fixed distance 'a' from the center, measured along the transverse axis. Knowing the vertices provides essential information for determining this distance and for positioning the hyperbola correctly on a coordinate plane.

Standard_Form_of_Hyperbola

The standard form of a hyperbola’s equation depends on whether its transverse axis is horizontal or vertical. For a vertical transverse axis, the form is (y - k)²/a² - (x - h)²/b² = 1, and for a horizontal transverse axis, it is (x - h)²/a² - (y - k)²/b² = 1. This form provides a systematic way to identify and manipulate the hyperbola's key features, such as its center, vertices, and asymptotic behavior.

Center_of_Hyperbola

The center of a hyperbola is the point (h, k) that lies midway between its vertices. It serves as the point of symmetry for the hyperbola and is used as the reference for constructing the graph, including plotting the vertices, foci, and asymptotes.

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