use vertices and asymptotes to graph each hyperbola. (x^2)/(16)-(y^2)/(25)=1

Name: Problem 6, Chapter 10: Introductory and Intermediate Algebra for College Students
Uploaded: 2020-08-03T11:11:29-08:00
Duration: 1 min 34 s
Channel: Joseph Lentino
Description: use vertices and asymptotes to graph each hyperbola. (x^2)/(16)-(y^2)/(25)=1

step 1 In this problem, I have x squared over 16 minus y squared over 25 equals 1. I know that this hyp

Use vertices and asymptotes to graph each hyperbola....

Key Concepts

Hyperbola

A hyperbola is a type of conic section defined as the set of all points in the plane for which the difference of the distances to two fixed points (the foci) is constant. The hyperbola consists of two separate branches, and its overall structure is determined by key parameters derived from its standard form equation.

Standard Form of a Hyperbola

The standard form of a hyperbola's equation makes it easier to extract important features such as the center, vertices, and asymptotes. For a hyperbola with a horizontal transverse axis, the equation is written as (x - h)²/a² - (y - k)²/b² = 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 relates to the distance used in determining the slopes of the asymptotes.

Vertices

Vertices are the points on a hyperbola that lie along the transverse axis and represent the closest points of each branch to the center. In the standard form equation, the coordinates of the vertices for a hyperbola with a horizontal transverse axis are given by (h ± a, k). These points are central to accurately graphing the shape and orientation of the hyperbola.

Asymptotes

Asymptotes are straight lines that define the directions along which the branches of a hyperbola extend as they stretch out to infinity. They serve as guidelines that the hyperbola approaches but never touches. For a hyperbola in standard form with a horizontal transverse axis, the equations of the asymptotes are y = k ± (b/a)(x - h), where the slopes ±(b/a) provide the directional angles that help in accurately sketching the hyperbola.

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