Find the equation for the hyperbola whose graph is shown – The equation of a hyperbola is a mathematical expression that describes the shape and position of a hyperbola on a graph. By understanding the equation of a hyperbola, we can determine its key features, such as its center, vertices, and asymptotes.
This guide provides a step-by-step process for finding the equation of a hyperbola from its graph, along with examples and applications.
Hyperbolas are conic sections that exhibit unique properties and have various real-world applications. In this guide, we will explore the equation of a hyperbola, its graphical representation, and how to determine the equation from a given graph. By understanding these concepts, we can gain a deeper understanding of hyperbolas and their significance in various fields.
Equation of a Hyperbola: Find The Equation For The Hyperbola Whose Graph Is Shown
A hyperbola is a conic section defined by the following equation:$$\frac(x-h)^2a^2-\frac(y-k)^2b^2=1$$where (h, k) is the center of the hyperbola, a is the distance from the center to the vertices, and b is the distance from the center to the co-vertices.
Horizontal Hyperbola, Find the equation for the hyperbola whose graph is shown
The equation of a horizontal hyperbola with center (h, k) and vertices (h ± a, k) is:$$\frac(x-h)^2a^2-\frac(y-k)^2b^2=1$$
Vertical Hyperbola
The equation of a vertical hyperbola with center (h, k) and vertices (h, k ± a) is:$$\frac(x-h)^2b^2-\frac(y-k)^2a^2=1$$
Graph of a Hyperbola
The graph of a hyperbola has the following key features:
- Two branches that extend indefinitely in opposite directions.
- Two vertices, one on each branch.
- Two asymptotes, one parallel to each axis.
- A center, which is the midpoint of the line segment connecting the vertices.
The asymptotes of a hyperbola are given by the equations:
- y = ±(b/a)(x
- h) for a horizontal hyperbola
- x = ±(a/b)(y
- k) for a vertical hyperbola
Determining the Equation from a Graph
To determine the equation of a hyperbola from its graph, follow these steps:
- Identify the center (h, k) as the midpoint of the line segment connecting the vertices.
- Determine whether the hyperbola is horizontal or vertical based on the orientation of the branches.
- Find the distance from the center to the vertices (a) and the co-vertices (b).
- Use the appropriate equation (horizontal or vertical) to write the equation of the hyperbola.
Examples and Applications
Examples:
Horizontal hyperbola
$\frac(x-3)^29-\frac(y-2)^24=1$
Vertical hyperbola
$\frac(x-1)^24-\frac(y+2)^29=1$ Applications:
- Hyperbolas are used in physics to model the trajectory of projectiles.
- Hyperbolas are used in engineering to design bridges and other structures.
Helpful Answers
What is the standard equation of a hyperbola?
The standard equation of a hyperbola is (x – h)^2/a^2 – (y – k)^2/b^2 = 1, where (h, k) is the center of the hyperbola, a is the distance from the center to the vertices along the transverse axis, and b is the distance from the center to the vertices along the conjugate axis.
How do I identify the center, vertices, and asymptotes of a hyperbola from its graph?
To identify the center, vertices, and asymptotes of a hyperbola from its graph, follow these steps: 1) Find the center of the hyperbola, which is the point of intersection of the transverse and conjugate axes. 2) The vertices are the points on the transverse axis that are equidistant from the center.
3) The asymptotes are the lines that the hyperbola approaches but never touches.
What are some real-world applications of hyperbolas?
Hyperbolas have various real-world applications, including: 1) In physics, hyperbolas are used to describe the trajectory of projectiles. 2) In engineering, hyperbolas are used to design bridges, arches, and other structures. 3) In astronomy, hyperbolas are used to calculate the orbits of comets and asteroids.
