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Chapter 13: Problem 33

Identify whether each equation, when graphed, will be a parabola, circle,ellipse, or hyperbola. Sketch the graph of each equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts. $$ \frac{y^{2}}{9}-\frac{x^{2}}{9}=1 $$

Short Answer

Expert verified
The equation represents a vertical hyperbola centered at the origin with vertices at \((0, 3)\) and \((0, -3)\), and asymptotes \(y = x\) and \(y = -x\).

Step by step solution

01

Recognize the Form of the Equation

The given equation is \( \frac{y^2}{9} - \frac{x^2}{9} = 1 \). This resembles the standard form of a hyperbola, which is \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \) or \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). Since the \( y^2 \) term is positive and the \( x^2 \) term is negative, this matches \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), indicating a vertical hyperbola.
02

Identify the Center

In a standard hyperbola equation, \( \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \), the center is at \((h,k)\). For \( \frac{y^2}{9} - \frac{x^2}{9} = 1 \), the center of the hyperbola is at the origin \((0,0)\), since the equation does not include \( h \) or \( k \).
03

Find the Vertices and Asymptotes

For the equation \( \frac{y^2}{9} - \frac{x^2}{9} = 1 \), the value of \( a^2 = 9 \), so \( a = 3 \). The vertices are \((0, 3)\) and \((0, -3)\). The asymptotes of a vertical hyperbola are given by \( y = \pm \frac{a}{b}x \). Here, \( \frac{a}{b} = \frac{3}{3} = 1 \), so the asymptotes are \( y = x \) and \( y = -x \).
04

Graph the Hyperbola

Begin by plotting the center at \((0, 0)\) on a coordinate plane. Then, plot the vertices at \((0, 3)\) and \((0, -3)\). Draw the asymptotes as diagonal lines through the origin with slopes of 1 and -1. Sketch the two branches of the hyperbola opening upwards and downwards, getting closer to but never touching the asymptotes.

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Key Concepts

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

Conic Sections
Conic sections are curves obtained by intersecting a plane with a cone. Depending on the angle and position of the intersection, you can get different types of curves:
- Circle
- Ellipse
- Parabola
- Hyperbola
Each of these has unique properties and equations. In our original problem, the equation given is for a hyperbola. Hyperbolas are one type of conic section that consists of two separate curves, known as branches. They occur when the intersecting plane is parallel to the cone's axis. Understanding these conic sections helps in identifying the kind of graph an equation represents.
Graphing Equations
Graphing equations involves plotting the curve that an equation represents on a coordinate plane. The equation \( \frac{y^2}{9} - \frac{x^2}{9} = 1 \) is in the form of a hyperbola, \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), indicating a vertical hyperbola.

To graph this hyperbola:
  • Start by plotting the center of the hyperbola, usually denoted by \((h, k)\). For this equation, it is \((0,0)\).
  • Next, identify the vertices, which are \((0, 3)\) and \((0, -3)\).
  • Draw the asymptotes through \((0,0)\) with slopes of 1 and -1.
  • Sketch the branches curving away from the center.
Graphing the equation visually represents its behavior, making abstract math problems tangible.
Vertex and Center Identification
In conic sections, the terms vertex and center appear frequently. For hyperbolas, identifying these points is essential for graphing and understanding their structure.

  • The **center** is the point where the hyperbola is balanced. In the equation \( \frac{y^2}{9} - \frac{x^2}{9} = 1 \), the lack of \(h\) and \(k\) signifies that the center is at the origin, \((0,0)\).
  • The **vertices** are key features. These points represent the intersection of the hyperbola with its axis of symmetry. In our equation, they are \((0, 3)\) and \((0, -3)\), since \(a = 3\).
Knowing how to find these points gives a clearer picture of the hyperbola's size and orientation.
Asymptotes
Asymptotes are straight lines that a curve approaches but never actually meets. In hyperbolas, they play a crucial role.

For a vertical hyperbola in the form \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), asymptotes provide a framework that guides the shape of the hyperbola's branches. They can be found using the formula \( y = \pm \frac{a}{b}x \).

In the provided equation \( \frac{y^2}{9} - \frac{x^2}{9} = 1 \), \(a = 3\) and \(b = 3\), giving us the asymptotes:
-\( y = x \)
-\( y = -x \)
Drawn through the origin \((0,0)\), they indicate the paths each branch of the hyperbola will follow, shaping its characteristic open curves.

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Most popular questions from this chapter

Graph each equation. $$ 4(x-1)^{2}+9(y+2)^{2}=36 $$

If you are given a list of equations of circles, parabolas, ellipses, and hyperbolas, explain how you could distinguish the different conic sections from their equations.

In 1893, Pittsburgh bridge builder George Ferris designed and built a gigantic revolving steel wheel whose height was 264 feet and diameter was 250 feet. This Ferris wheel opened at the 1893 exposition in Chicago. It had 36 wooden cars, each capable of holding 60 passengers. (Source: The Handy Science Answer Book) a. What was the radius of this Ferris wheel? b. How close is the wheel to the ground? C. How high is the center of the wheel from the ground? d. Using the axes in the drawing, what are the coordinates of the center of the wheel? e. Use parts a and \(\mathbf{d}\) to write an equation of the wheel.

The graph of each equation is a parabola. Find the vertex of the parabola and then graph it. See Examples 1 through 4. $$ y=x^{2}+10 x+20 $$

Although there are many larger observation wheels on the horizon, as of this writing the largest observation wheel in the world is the Singapore Flyer. From the Flyer, you can see up to \(45 \mathrm{~km}\) away. Each of the 28 enclosed capsules holds 28 passengers, completes a full rotation every 32 minutes. Its diameter is 150 meters, and the height of this giant wheel is 165 meters. (Source: singaporeflyer.com) a. What is the radius of the Singapore Flyer? b. How close is the wheel to the ground? C. How high is the center of the wheel from the ground? d. Using the axes in the drawing, what are the coordinates of the center of the wheel? e. Use parts a and \(\mathbf{d}\) to write an equation of the Singapore Flyer.
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