**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 WaveThe 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 FunctionThe 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 $.

Use the vertical line test to determine if the graph represents a function. Look at the graph to…

Answered: Select all of the following graphs…

Select all of the following graphs which represent y as a function of æ. -5 4 3 -2 -2 -3 -4 -3 -2- 3 5+ 2. -5 -4 -3 -2 to

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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Concept explainers

Vertices Of An Ellipse

In the study of the conic section of the geometry, An ellipse is a plane curve surrounding two focal points, or foci where the set of all locus of points in the plane has their sum of the distances to the two foci is constant. On the ellipse curve, it is the intersection point of the main axis. In an ellipse graph, the center of the axis is the midpoint of the line segment connecting the foci. The major axis is the line section that runs through the ellipse's foci, while the minor axis runs through the middle and is perpendicular to the major axis.. The ellipse has two endpoints on the major axis which are known as the vertices (in plural), vertex in the singular. The equation of the major axis is 2a, the equation of the minor is 2b and the distance between the major axis and minor axis is 2c. The semi-major axis has a length of a and the semi-minor axis has a length of b. The ellipse's vertices are determined by the elliptical orbit's equation and graph.

Question

**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 $.

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