### Analyzing Graphs: Understanding Trigonometric FunctionsThis page provides an analysis of two distinct graphs for educational purposes. Each graph will be explained in detail regarding its characteristics and what it represents.#### Graph 1: Sine Wave**Description**: - The first graph is a sine wave, a periodic function. - The curve oscillates above and below the x-axis, indicating periodic behavior.- The x-axis represents the input angle in radians, ranging from -5 to 5.- The y-axis represents the sine of the input angle, ranging from -5 to 5.- Key points include intersections at (-3π, -4), (-π, -2), (π,2) and (3π, 4), peaks at -4 and 4, and troughs at -2 and 2.**Characteristics**:- **Periodicity**: The sine wave repeats every 2π units.- **Symmetry**: The graph is symmetric with respect to the origin, showcasing an odd function.- **Amplitudes**: The highest point (peak) and lowest point (trough) are equidistant from the x-axis, indicating the amplitude of the wave.#### Graph 2: Constant Function**Description**: - The second graph is a horizontal line, indicating a constant function.- The line is parallel to the x-axis.- The function value does not change with varying x-values.- Both axes range from -5 to 5.- The horizontal line likely represents a function with a constant value close to 0.**Characteristics**:- **Zero Slope**: The graph's slope is zero, indicating no change in y-value with respect to x.- **Constant Value**: All points on the line have the same y-coordinate.### Educational Insights- Comparing these two graphs helps in understanding the difference between a periodic function (like the sine wave) and a constant function. - This distinction is critical in various fields such as physics, engineering, and mathematics, where understanding the behavior of different functions is fundamental.- Observing the sine wave helps students comprehend concepts such as amplitude, period, and phase shift.- The constant function graph is helpful in understanding the basic concept of slope and intercept in linear functions.These visual representations are an excellent starting point for further exploration into more complex trigonometric and algebraic functions. **Title: Identifying Functions from Graphs****Instructions:**Select all of the following graphs which represent $ y $ as a function of $ x $.**Graph Descriptions:**1. **First Graph:** - This graph depicts a straight blue line passing through the origin (0, 0) and extending diagonally in a positive linear direction. The line crosses each axis at equal intervals, implying it represents a linear function like $ y = x $. - The $ x $-axis ranges from -5 to 5, and the $ y $-axis ranges from -5 to 5. - This graph represents $ y $ as a function of $ x $.2. **Second Graph:** - This graph shows a blue circle centered at (-2, 2) with a radius of 1 unit. - Similar to the first graph, the $ x $-axis ranges from -5 to 5, and the $ y $-axis ranges from -5 to 5. - This graph does not represent $ y $ as a function of $ x $ because for some values of $ x $, there are multiple corresponding $ y $ values.3. **Third Graph:** - This graph presents a blue curve which has multiple peaks and valleys, exhibiting a polynomial-like behavior. It crosses the $ y $-axis at a single point but intersects the $ x $-axis multiple times and appears to have vertical asymptotes. - The $ x $-axis and the $ y $-axis both range from -5 to 5. - This graph represents $ y $ as a function of $ x $.4. **Fourth Graph:** - This graph illustrates a blue curve that appears to loop back on itself. It resembles a function that does not pass the vertical line test, meaning that for some $ x $ values, there are multiple corresponding $ y $ values. - The $ x $-axis and the $ y $-axis both range from -5 to 5. - This graph does not represent $ y $ as a function of $ x $.**Conclusion:**- **Graphs 1 and 3** represent $ y $ as a function of $ x $.- **Graphs 2 and 4** do not represent $ y $ as a function of \( x \

Vertical line test: A graph represents a function if there are no vertical lines that intersect the…

Answered: Select all of the following graphs…

Select all of the following graphs which represent y as a function of r. 4- -5 -4 -3 -2 - -2+ -3 -4 4 -3 -2 -1 -4- -5 -4 -3 2 -2 -4 4- 2. -5 -4 -3 -4

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter1: Expressions And Functions

Section: Chapter Questions

Problem 74SGR

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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

### Analyzing Graphs: Understanding Trigonometric Functions

This page provides an analysis of two distinct graphs for educational purposes. Each graph will be explained in detail regarding its characteristics and what it represents.

#### Graph 1: Sine Wave
**Description**:
- The first graph is a sine wave, a periodic function.
- The curve oscillates above and below the x-axis, indicating periodic behavior.
- The x-axis represents the input angle in radians, ranging from -5 to 5.
- The y-axis represents the sine of the input angle, ranging from -5 to 5.
- Key points include intersections at (-3π, -4), (-π, -2), (π,2) and (3π, 4), peaks at -4 and 4, and troughs at -2 and 2.

**Characteristics**:
- **Periodicity**: The sine wave repeats every 2π units.
- **Symmetry**: The graph is symmetric with respect to the origin, showcasing an odd function.
- **Amplitudes**: The highest point (peak) and lowest point (trough) are equidistant from the x-axis, indicating the amplitude of the wave.

#### Graph 2: Constant Function
**Description**:
- The second graph is a horizontal line, indicating a constant function.
- The line is parallel to the x-axis.
- The function value does not change with varying x-values.
- Both axes range from -5 to 5.
- The horizontal line likely represents a function with a constant value close to 0.

**Characteristics**:
- **Zero Slope**: The graph's slope is zero, indicating no change in y-value with respect to x.
- **Constant Value**: All points on the line have the same y-coordinate.

### Educational Insights
- Comparing these two graphs helps in understanding the difference between a periodic function (like the sine wave) and a constant function.
- This distinction is critical in various fields such as physics, engineering, and mathematics, where understanding the behavior of different functions is fundamental.
- Observing the sine wave helps students comprehend concepts such as amplitude, period, and phase shift.
- The constant function graph is helpful in understanding the basic concept of slope and intercept in linear functions.

These visual representations are an excellent starting point for further exploration into more complex trigonometric and algebraic functions.

**Title: Identifying Functions from Graphs**

**Instructions:**
Select all of the following graphs which represent $ y $ as a function of $ x $.

**Graph Descriptions:**

1. **First Graph:**
- This graph depicts a straight blue line passing through the origin (0, 0) and extending diagonally in a positive linear direction. The line crosses each axis at equal intervals, implying it represents a linear function like $ y = x $.
- The $ x $-axis ranges from -5 to 5, and the $ y $-axis ranges from -5 to 5.
- This graph represents $ y $ as a function of $ x $.

2. **Second Graph:**
- This graph shows a blue circle centered at (-2, 2) with a radius of 1 unit.
- Similar to the first graph, the $ x $-axis ranges from -5 to 5, and the $ y $-axis ranges from -5 to 5.
- This graph does not represent $ y $ as a function of $ x $ because for some values of $ x $, there are multiple corresponding $ y $ values.

3. **Third Graph:**
- This graph presents a blue curve which has multiple peaks and valleys, exhibiting a polynomial-like behavior. It crosses the $ y $-axis at a single point but intersects the $ x $-axis multiple times and appears to have vertical asymptotes.
- The $ x $-axis and the $ y $-axis both range from -5 to 5.
- This graph represents $ y $ as a function of $ x $.

4. **Fourth Graph:**
- This graph illustrates a blue curve that appears to loop back on itself. It resembles a function that does not pass the vertical line test, meaning that for some $ x $ values, there are multiple corresponding $ y $ values.
- The $ x $-axis and the $ y $-axis both range from -5 to 5.
- This graph does not represent $ y $ as a function of $ x $.

**Conclusion:**
- **Graphs 1 and 3** represent $ y $ as a function of $ x $.
- **Graphs 2 and 4** do not represent $ y $ as a function of \( x \

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