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,...
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![**Understanding Graphs of Mathematical Functions**
On this educational page, we will explore and analyze two different graphs typically encountered in trigonometry and linear algebra.
### Graph 1: Sine Wave
The first graph represents the function \( f(x) = \sin(x) \). This sine wave is a fundamental concept in trigonometry and depicts periodic oscillation. Key features of the sine wave include:
- **Periodicity**: The graph repeats every \( 2\pi \) units along the x-axis.
- **Amplitude**: The maximum and minimum values of the sine function are 1 and -1 respectively, indicating the amplitude of oscillation.
- **Zero Crossings**: The function crosses the x-axis at integer multiples of \( \pi \), where the sine value is zero.
- **Symmetry**: The graph is symmetric about the origin, indicating it is an odd function, i.e., \( \sin(-x) = -\sin(x) \).
### Graph 2: Constant Function
The second graph shows a horizontal line, representing a constant function \( g(x) = k \) where \( k \) is a constant. In this particular graph:
- **Consistency**: The value of \( k \) depicted is 0, hence the function is \( g(x) = 0 \).
- **Horizontal Line**: This line runs parallel to the x-axis and indicates that for every value of \( x \), \( g(x) \) remains equal to 0.
- **Zero Slope**: Since it is a constant function, its slope is zero, representing no change over the x-axis.
These fundamental graphs provide the groundwork for more complex mathematical analysis and are essential for both theoretical understanding and practical applications in various fields of science and engineering.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F633100ea-0428-4ac9-9895-e60fbf23e986%2Fade0418c-9c1a-4277-b50d-be32795ab360%2Fmyjl6ib.png&w=3840&q=75)
Transcribed Image Text:**Understanding Graphs of Mathematical Functions**
On this educational page, we will explore and analyze two different graphs typically encountered in trigonometry and linear algebra.
### Graph 1: Sine Wave
The first graph represents the function \( f(x) = \sin(x) \). This sine wave is a fundamental concept in trigonometry and depicts periodic oscillation. Key features of the sine wave include:
- **Periodicity**: The graph repeats every \( 2\pi \) units along the x-axis.
- **Amplitude**: The maximum and minimum values of the sine function are 1 and -1 respectively, indicating the amplitude of oscillation.
- **Zero Crossings**: The function crosses the x-axis at integer multiples of \( \pi \), where the sine value is zero.
- **Symmetry**: The graph is symmetric about the origin, indicating it is an odd function, i.e., \( \sin(-x) = -\sin(x) \).
### Graph 2: Constant Function
The second graph shows a horizontal line, representing a constant function \( g(x) = k \) where \( k \) is a constant. In this particular graph:
- **Consistency**: The value of \( k \) depicted is 0, hence the function is \( g(x) = 0 \).
- **Horizontal Line**: This line runs parallel to the x-axis and indicates that for every value of \( x \), \( g(x) \) remains equal to 0.
- **Zero Slope**: Since it is a constant function, its slope is zero, representing no change over the x-axis.
These fundamental graphs provide the groundwork for more complex mathematical analysis and are essential for both theoretical understanding and practical applications in various fields of science and engineering.
![**Title: Identifying Functions from Graphs**
**Objective**: Learn how to determine if a graph represents \( y \) as a function of \( x \).
**Instructions**: Select all the following graphs which represent \( y \) as a function of \( x \).
### Graph Analysis:
1. **Graph 1**: A straight line extending from the bottom left to the top right of the graph.
- **Description**: The graph passes the vertical line test, as any vertical line would only intersect the curve at one point.
- **Function?**: Yes.
2. **Graph 2**: A circle centered at (-1, 1) with a radius of 2.
- **Description**: The graph fails the vertical line test, as some vertical lines will intersect the circle at two points.
- **Function?**: No.
3. **Graph 3**: A polynomial curve resembling an "N" shape, starting from the bottom left, rising, then dipping, and finally rising steeply to the top.
- **Description**: The graph passes the vertical line test, as any vertical line would only intersect the curve at one point.
- **Function?**: Yes.
4. **Graph 4**: A complex wave-like curve with multiple peaks and valleys.
- **Description**: The graph fails the vertical line test in some regions as some vertical lines intersect the curve at multiple points.
- **Function?**: No.
### Conclusion:
Based on the vertical line test, **Graphs 1 and 3** represent \( y \) as a function of \( x \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F633100ea-0428-4ac9-9895-e60fbf23e986%2Fade0418c-9c1a-4277-b50d-be32795ab360%2Fvi1jjnu.png&w=3840&q=75)
Transcribed Image Text:**Title: Identifying Functions from Graphs**
**Objective**: Learn how to determine if a graph represents \( y \) as a function of \( x \).
**Instructions**: Select all the following graphs which represent \( y \) as a function of \( x \).
### Graph Analysis:
1. **Graph 1**: A straight line extending from the bottom left to the top right of the graph.
- **Description**: The graph passes the vertical line test, as any vertical line would only intersect the curve at one point.
- **Function?**: Yes.
2. **Graph 2**: A circle centered at (-1, 1) with a radius of 2.
- **Description**: The graph fails the vertical line test, as some vertical lines will intersect the circle at two points.
- **Function?**: No.
3. **Graph 3**: A polynomial curve resembling an "N" shape, starting from the bottom left, rising, then dipping, and finally rising steeply to the top.
- **Description**: The graph passes the vertical line test, as any vertical line would only intersect the curve at one point.
- **Function?**: Yes.
4. **Graph 4**: A complex wave-like curve with multiple peaks and valleys.
- **Description**: The graph fails the vertical line test in some regions as some vertical lines intersect the curve at multiple points.
- **Function?**: No.
### Conclusion:
Based on the vertical line test, **Graphs 1 and 3** represent \( y \) as a function of \( x \).
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