Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
Which graph corresponds to the equation y^2/49 - x^2/45 = 1

Transcribed Image Text:### Identifying Hyperbolas
#### Question:
Which graph corresponds to the equation \(\frac{y^2}{49} - \frac{x^2}{25} = 1\)?
#### Explanation:
This equation represents the equation of a hyperbola. Notice the vertices, asymptotes, and orientation.
**Analysis of the Graphs:**
1. **First Graph**:
- Orientation: Horizontal hyperbola.
- Equation Format: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).
- Characteristics: Opens left and right.
2. **Second Graph**:
- Orientation: Vertical hyperbola.
- Equation Format: \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\).
- Characteristics: Opens up and down.
### Matching the Equation:
The given equation \(\frac{y^2}{49} - \frac{x^2}{25} = 1\) matches the form of a vertical hyperbola.
**Conclusion**:
The correct graph corresponding to the given equation is the **second graph**.
### Detailed Graph Description:
**Second Graph**:
- The hyperbola opens upwards and downwards.
- The vertices are positioned at \((0, \pm7)\).
- The asymptotes intersect at the origin (0, 0). The slopes of the asymptotes are determined by the coefficients under \(x^2\) and \(y^2\) (i.e., \(\pm\frac{7}{5}\)).
### Learning Objective:
Understand the identification and characteristics of hyperbolas and how their graphs correspond to their equations.
![### Understanding Graphs of Specific Functions
#### Introduction
In this section, we will study and analyze the shapes and characteristics of several mathematical functions as represented on a Cartesian plane. This will aid in understanding their behaviors and applications in various fields.
#### Graph 1: Parabolic Curve (Upward Opening)
The first graph displays a parabolic curve opening upwards. This is typical of a quadratic function of the form:
\[ y = ax^2 \]
where \( a \) is a positive constant. The vertex of the parabola is at the origin (0,0), indicating this is a standard parabolic curve without horizontal or vertical shifts.
**Characteristics:**
- Symmetric about the y-axis.
- Minimum point at the origin.
- As \( x \) increases or decreases, \( y \) increases.
#### Graph 2: Inverted Parabolic Curve
The second graph depicts an inverted parabolic curve, which graphs a quadratic function of the form:
\[ y = -ax^2 \]
where \( a \) is a positive constant. The vertex, like the first parabola, is at the origin (0,0).
**Characteristics:**
- Symmetric about the y-axis.
- Maximum point at the origin.
- As \( x \) increases or decreases, \( y \) decreases.
#### Graph 3: Hyperbolic Curve
The third graph showcases a hyperbolic curve representing a hyperbola of the form:
\[ xy = c \]
or a standard hyperbola such as:
\[ y = \frac{a}{x} \]
where \( a \) and \( c \) are constants. The graph is characterized by two symmetrical curved branches.
**Characteristics:**
- Asymptotes along the coordinate axes (x=0 and y=0).
- The branches head towards infinity as \( x \) increases or decreases.
- Not symmetric about the y-axis or x-axis but has reflective symmetry about the origin.
### Conclusion
By interpreting and graphing these functions, one can gain a hands-on understanding of their geometric properties and mathematical implications. It’s crucial to be able to visualize these graphs to predict function behavior in practical and theoretical applications.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe20fc6e6-2ea7-4c1e-8366-712724b6961a%2F6993bd43-71fb-4cec-8431-dceccc873561%2Fbytcvd_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Graphs of Specific Functions
#### Introduction
In this section, we will study and analyze the shapes and characteristics of several mathematical functions as represented on a Cartesian plane. This will aid in understanding their behaviors and applications in various fields.
#### Graph 1: Parabolic Curve (Upward Opening)
The first graph displays a parabolic curve opening upwards. This is typical of a quadratic function of the form:
\[ y = ax^2 \]
where \( a \) is a positive constant. The vertex of the parabola is at the origin (0,0), indicating this is a standard parabolic curve without horizontal or vertical shifts.
**Characteristics:**
- Symmetric about the y-axis.
- Minimum point at the origin.
- As \( x \) increases or decreases, \( y \) increases.
#### Graph 2: Inverted Parabolic Curve
The second graph depicts an inverted parabolic curve, which graphs a quadratic function of the form:
\[ y = -ax^2 \]
where \( a \) is a positive constant. The vertex, like the first parabola, is at the origin (0,0).
**Characteristics:**
- Symmetric about the y-axis.
- Maximum point at the origin.
- As \( x \) increases or decreases, \( y \) decreases.
#### Graph 3: Hyperbolic Curve
The third graph showcases a hyperbolic curve representing a hyperbola of the form:
\[ xy = c \]
or a standard hyperbola such as:
\[ y = \frac{a}{x} \]
where \( a \) and \( c \) are constants. The graph is characterized by two symmetrical curved branches.
**Characteristics:**
- Asymptotes along the coordinate axes (x=0 and y=0).
- The branches head towards infinity as \( x \) increases or decreases.
- Not symmetric about the y-axis or x-axis but has reflective symmetry about the origin.
### Conclusion
By interpreting and graphing these functions, one can gain a hands-on understanding of their geometric properties and mathematical implications. It’s crucial to be able to visualize these graphs to predict function behavior in practical and theoretical applications.
Expert Solution

This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images

Recommended textbooks for you

Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON

Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning

Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning

Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON

Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning

Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning

Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON

Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press

College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education