Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Concept explainers
Transformation of Graphs
The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.
Exponential Functions
The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, concepts, structures, graphs, interpreter series, work formulas, and examples of functions involving exponents.
Question
F:# 18
![### Graphing Quadratic Equations
#### Problem 18:
Graph the following equation:
\[ y^2 + 12 = 3x^2 \]
##### Steps to Graph:
1. **Rewrite the equation for simplicity**:
\[ y^2 + 12 = 3x^2 \]
\[ y^2 = 3x^2 - 12 \]
\[ \frac{y^2}{12} = \frac{3x^2}{12} - 1 \]
\[ \frac{y^2}{12} = \frac{x^2}{4} - 1 \]
\[ \frac{x^2}{4} - \frac{y^2}{12} = 1 \]
2. **Identify the type of conic section**:
The equation \(\frac{x^2}{4} - \frac{y^2}{12} = 1\) represents a hyperbola centered at the origin with the transverse axis along the x-axis.
3. **Determine the vertices and asymptotes**:
- **Vertices**: Since \( a^2 = 4 \), the vertices are at \(( \pm 2, 0)\).
- **Asymptotes**: The equations of the asymptotes are determined by setting the equation of the hyperbola equal to zero:
\[\frac{x^2}{4} - \frac{y^2}{12} = 0 \]
Simplifying gives the asymptotes:
\[ \frac{y^2}{12} = \frac{x^2}{4} \]
\[ y = \pm \sqrt{3} x \]
To graph the given equation:
1. Plot the vertices at \((2, 0)\) and \((-2, 0)\).
2. Draw the asymptotes, which are straight lines passing through the origin with slopes \(\sqrt{3}\) and \(-\sqrt{3}\).
3. Sketch the hyperbola opening left and right from the vertices, approaching the asymptotes.
This visual representation helps in understanding the structure and features of hyperbolas, aiding in analysis and problem-solving with conic sections.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c34de7e-5935-4303-bb64-ce025f7422a2%2F53366eda-29d4-4a78-9de0-0e9739350feb%2Fglw2y3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Graphing Quadratic Equations
#### Problem 18:
Graph the following equation:
\[ y^2 + 12 = 3x^2 \]
##### Steps to Graph:
1. **Rewrite the equation for simplicity**:
\[ y^2 + 12 = 3x^2 \]
\[ y^2 = 3x^2 - 12 \]
\[ \frac{y^2}{12} = \frac{3x^2}{12} - 1 \]
\[ \frac{y^2}{12} = \frac{x^2}{4} - 1 \]
\[ \frac{x^2}{4} - \frac{y^2}{12} = 1 \]
2. **Identify the type of conic section**:
The equation \(\frac{x^2}{4} - \frac{y^2}{12} = 1\) represents a hyperbola centered at the origin with the transverse axis along the x-axis.
3. **Determine the vertices and asymptotes**:
- **Vertices**: Since \( a^2 = 4 \), the vertices are at \(( \pm 2, 0)\).
- **Asymptotes**: The equations of the asymptotes are determined by setting the equation of the hyperbola equal to zero:
\[\frac{x^2}{4} - \frac{y^2}{12} = 0 \]
Simplifying gives the asymptotes:
\[ \frac{y^2}{12} = \frac{x^2}{4} \]
\[ y = \pm \sqrt{3} x \]
To graph the given equation:
1. Plot the vertices at \((2, 0)\) and \((-2, 0)\).
2. Draw the asymptotes, which are straight lines passing through the origin with slopes \(\sqrt{3}\) and \(-\sqrt{3}\).
3. Sketch the hyperbola opening left and right from the vertices, approaching the asymptotes.
This visual representation helps in understanding the structure and features of hyperbolas, aiding in analysis and problem-solving with conic sections.
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
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning