/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 11 Graph each hyperbola. $$ \fr... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 11

Graph each hyperbola. $$ \frac{x^{2}}{25}-\frac{y^{2}}{36}=1 $$

Short Answer

Expert verified
Vertices at \((5, 0)\), \((-5, 0)\); asymptotes \(y = \pm \frac{6}{5}x\).

Step by step solution

01

Identify the Standard Form

The given hyperbola is in the standard form: \[\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\]. Here, \(a^2 = 25\) and \(b^2 = 36\).
02

Calculate the Values of a and b

From the equation, \(a^2 = 25\) gives \(a = 5\), and \(b^2 = 36\) gives \(b = 6\).
03

Determine the Center of the Hyperbola

For the hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\), the center is at the origin, \((0, 0)\).
04

Find the Vertices

The vertices are located at \((\pm a, 0)\). Using the value of \(a = 5\), the vertices are \((5, 0)\) and \((-5, 0)\).
05

Find the Asymptotes

For the hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\), the equations of the asymptotes are \[y = \pm \frac{b}{a}x\]. Since \(\frac{b}{a} = \frac{6}{5}\), the asymptotes are \[y = \pm \frac{6}{5}x\].
06

Sketch the Hyperbola

1. Plot the center at \((0, 0)\).2. Plot the vertices at \((5, 0)\) and \((-5, 0)\).3. Draw the asymptotes \(y = \pm \frac{6}{5}x\).4. Sketch the branches opening left and right through the vertices and approaching the asymptotes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hyperbola Standard Form
The standard form of a hyperbola is crucial for understanding its properties and graphing it correctly. A hyperbola's equation typically appears as either \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) or \(\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1\). The first form opens left and right, while the second opens up and down. The constants \(a^{2}\) and \(b^{2}\) represent the squares of distances related to the hyperbola. Specifically, \(a\) is the distance from the center to the vertices along the transverse axis.
Hyperbola Vertices
The vertices of a hyperbola are key points on its graph. They lie along the transverse axis. In the equation \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\), the vertices are located at \((\text{卤}a, 0)\). To find the specific vertices, identify \(\text{卤}a\) by solving for \(a\). For example, if \(a^{2} = 25\), then \(a = 5\). The vertices are therefore \((5, 0)\) and \((-5, 0)\). These points mark where the branches of the hyperbola cross the transverse axis.
Hyperbola Asymptotes
Asymptotes are lines that the hyperbola approaches but never touches. They provide a guide for sketching the curve. For the hyperbola in the form \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\), the equations of the asymptotes are \(y = \text{卤} \frac{b}{a}x\). Calculate \( \frac{b}{a} \) using the given \( a \) and \( b \) values. For example, if \( a = 5 \) and \( b = 6 \), then \( \frac{b}{a} = \frac{6}{5}\). The asymptotes are therefore \( y = \text{卤} \frac{6}{5}x \). These lines help define the direction in which the branches of the hyperbola open.
Hyperbola Center
The center of a hyperbola is the starting point for plotting. It is where the two axes (transverse and conjugate) intersect. For a hyperbola in the form \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\), the center is at the origin, \((0, 0)\). To graph a hyperbola, start by plotting this center point. The values of \(a\) and \(b\) determine the distances from the center to the vertices and the shape of the asymptotes.

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