We will consider two cases: those that are centered at the origin, and those that are centered at a point other than the origin. The difference of the distances from the foci to the vertex is. The derivation of the equation of a hyperbola is based on applying the distance formula , but is again beyond the scope of this text. When we are given the equation of a hyperbola, we can use this relationship to identify its vertices and foci.
The hyperbola is centered at the origin, so the vertices serve as the y -intercepts of the graph. Just as with ellipses, writing the equation for a hyperbola in standard form allows us to calculate the key features: its center, vertices, co-vertices, foci, asymptotes, and the lengths and positions of the transverse and conjugate axes. Conversely, an equation for a hyperbola can be found given its key features.
We begin by finding standard equations for hyperbolas centered at the origin. Then we will turn our attention to finding standard equations for hyperbolas centered at some point other than the origin. This relationship is used to write the equation for a hyperbola when given the coordinates of its foci and vertices.
The vertices and foci are on the x -axis. Like the graphs for other equations, the graph of a hyperbola can be translated. The asymptotes of the hyperbola coincide with the diagonals of the central rectangle.
We can use this relationship along with the midpoint and distance formulas to find the standard equation of a hyperbola when the vertices and foci are given.
The y -coordinates of the vertices and foci are the same, so the transverse axis is parallel to the x -axis. Thus, the equation of the hyperbola will have the form. Applying the midpoint formula, we have. As we discussed at the beginning of this section, hyperbolas have real-world applications in many fields, such as astronomy, physics, engineering, and architecture.
The design efficiency of hyperbolic cooling towers is particularly interesting. Cooling towers are used to transfer waste heat to the atmosphere and are often touted for their ability to generate power efficiently.
Because of their hyperbolic form, these structures are able to withstand extreme winds while requiring less material than any other forms of their size and strength. For example a foot tower can be made of a reinforced concrete shell only 6 or 8 inches wide! The first hyperbolic towers were designed in and were 35 meters high.
Today, the tallest cooling towers are in France, standing a remarkable meters tall. In addition, a hyperbola is formed by the intersection of a cone with an oblique plane that intersects the base.
It consists of two separate curves, called branches The two separate curves of a hyperbola. Points on the separate branches of the graph where the distance is at a minimum are called vertices. Points on the separate branches of a hyperbola where the distance is a minimum. Unlike a parabola, a hyperbola is asymptotic to certain lines drawn through the center.
In this section, we will focus on graphing hyperbolas that open left and right or upward and downward. The asymptotes are drawn dashed as they are not part of the graph; they simply indicate the end behavior of the graph. The center is h , k , a defines the transverse axis, and b defines the conjugate axis.
The center is h , k , b defines the transverse axis, and a defines the conjugate axis. The asymptotes are essential for determining the shape of any hyperbola.
To easily sketch the asymptotes we make use of two special line segments through the center using a and b. Given any hyperbola, the transverse axis The line segment formed by the vertices of a hyperbola.
The conjugate axis A line segment through the center of a hyperbola that is perpendicular to the transverse axis. The rectangle defined by the transverse and conjugate axes is called the fundamental rectangle The rectangle formed using the endpoints of a hyperbolas, transverse and conjugate axes. These lines are the asymptotes that define the shape of the hyperbola. Therefore, given standard form, many of the properties of a hyperbola are apparent. The graph of a hyperbola is completely determined by its center, vertices, and asymptotes.
In this case, the expression involving x has a positive leading coefficient; therefore, the hyperbola opens left and right. From the center 5 , 4 , mark points 3 units left and right as well as 2 units up and down.
Connect these points with a rectangle as follows:. The lines through the corners of this rectangle define the asymptotes. Use these dashed lines as a guide to graph the hyperbola opening left and right passing through the vertices. In this case, the expression involving y has a positive leading coefficient; therefore, the hyperbola opens upward and downward.
Connect these points with a rectangle. Use these dashed lines as a guide to graph the hyperbola opening upward and downward passing through the vertices. Note : When given a hyperbola opening upward and downward, as in the previous example, it is a common error to interchange the values for the center, h and k. This is the case because the quantity involving the variable y usually appears first in standard form.
Take care to ensure that the y -value of the center comes from the quantity involving the variable y and that the x -value of the center is obtained from the quantity involving the variable x. As with any graph, we are interested in finding the x - and y -intercepts. Take a moment to compare these to the sketch of the graph in the previous example. Consider the hyperbola centered at the origin,.
If e is close to one, the branches of the hyperbola are very narrow, but if e is much greater than one, then the branches of the hyperbola are very flat. SparkTeach Teacher's Handbook. Summary Hyperbolas. Previous section Problems Next section Problems. Take a Study Break.
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