Explain How The Graph Of Y = − 2 Sin ⁡ ( X ) − 1 Y = -2 \sin(x) - 1 Y = − 2 Sin ( X ) − 1 Differs From The Parent Function. Make Sure To Compare Key Points Used For Graphing.

Introduction

The graph of a function is a visual representation of its behavior, and understanding how different functions relate to each other is crucial in mathematics. In this article, we will explore the graph of $y = -2 \sin(x) - 1$ and compare it to its parent function, $y = \sin(x)$. We will examine the key points used for graphing and discuss how the transformations affect the graph.

Understanding the Parent Function

The parent function of $y = -2 \sin(x) - 1$ is $y = \sin(x)$. The graph of the sine function is a periodic curve that oscillates between -1 and 1. It has a period of $2\pi$, which means it repeats every $2\pi$ units. The graph of the sine function has a maximum value of 1 at $x = \frac{\pi}{2}$ and a minimum value of -1 at $x = \frac{3\pi}{2}$.

Transformations of the Parent Function

The graph of $y = -2 \sin(x) - 1$ is a transformation of the parent function $y = \sin(x)$. There are three main transformations that occur:

1. Vertical Stretch: The graph of $y = -2 \sin(x) - 1$ is a vertical stretch of the parent function by a factor of 2. This means that the amplitude of the graph is doubled, resulting in a more extreme oscillation.
2. Reflection: The graph of $y = -2 \sin(x) - 1$ is a reflection of the parent function across the x-axis. This means that the graph is flipped upside down, resulting in a negative value for the function.
3. Vertical Shift: The graph of $y = -2 \sin(x) - 1$ is a vertical shift of the parent function down by 1 unit. This means that the graph is shifted down by 1 unit, resulting in a lower minimum value.

Key Points Used for Graphing

To graph the function $y = -2 \sin(x) - 1$, we need to identify the key points used for graphing. These key points include:

• Period: The period of the graph is still $2\pi$, which means it repeats every $2\pi$ units.
• Amplitude: The amplitude of the graph is 2, which means it oscillates between -2 and 2.
• Maximum Value: The maximum value of the graph is -2, which occurs at $x = \frac{\pi}{2}$.
• Minimum Value: The minimum value of the graph is -3, which occurs at $x = \frac{3\pi}{2}$.

Comparing Key Points

To compare the key points of the graph of $y = -2 \sin(x) - 1$ to its parent function, we can examine the following:

• Period: The period of both graphs is the same, $2\pi$.
• Amplitude: The amplitude of the graph of $y = -2 \sin(x) - 1$ is 2, while the amplitude of the parent function is 1.
• Maximum Value: The maximum value of the graph of $y = -2 \sin(x) - 1$ is -2, while the maximum value of the parent function is 1.
• Minimum Value: The minimum value of the graph of $y = -2 \sin(x) - 1$ is -3, while the minimum value of the parent function is -1.

Conclusion

In conclusion, the graph of $y = -2 \sin(x) - 1$ differs from its parent function $y = \sin(x)$ in several key ways. The graph is a vertical stretch of the parent function by a factor of 2, a reflection across the x-axis, and a vertical shift down by 1 unit. The key points used for graphing, including period, amplitude, maximum value, and minimum value, are also affected by these transformations. Understanding these transformations and key points is crucial in graphing and analyzing functions in mathematics.

References

• [1] "Graphing Sine and Cosine Functions" by Paul Dawkins
• [2] "Transformations of Functions" by Math Open Reference
• [3] "Graphing Functions" by Khan Academy

Additional Resources

• [1] "Graphing Sine and Cosine Functions" by Paul Dawkins (video)
• [2] "Transformations of Functions" by Math Open Reference (interactive graphing tool)
• [3] "Graphing Functions" by Khan Academy (practice problems and quizzes)
  Q&A: Explaining the Graph of $y = -2 \sin(x) - 1$ and its Differences from the Parent Function =====================================================================================

Frequently Asked Questions

Q: What is the parent function of $y = -2 \sin(x) - 1$?

A: The parent function of $y = -2 \sin(x) - 1$ is $y = \sin(x)$. The graph of the sine function is a periodic curve that oscillates between -1 and 1.

Q: What are the three main transformations that occur in the graph of $y = -2 \sin(x) - 1$?

A: The three main transformations that occur in the graph of $y = -2 \sin(x) - 1$ are:

1. Vertical Stretch: The graph of $y = -2 \sin(x) - 1$ is a vertical stretch of the parent function by a factor of 2.
2. Reflection: The graph of $y = -2 \sin(x) - 1$ is a reflection of the parent function across the x-axis.
3. Vertical Shift: The graph of $y = -2 \sin(x) - 1$ is a vertical shift of the parent function down by 1 unit.

Q: What is the period of the graph of $y = -2 \sin(x) - 1$?

A: The period of the graph of $y = -2 \sin(x) - 1$ is still $2\pi$, which means it repeats every $2\pi$ units.

Q: What is the amplitude of the graph of $y = -2 \sin(x) - 1$?

A: The amplitude of the graph of $y = -2 \sin(x) - 1$ is 2, which means it oscillates between -2 and 2.

Q: What is the maximum value of the graph of $y = -2 \sin(x) - 1$?

A: The maximum value of the graph of $y = -2 \sin(x) - 1$ is -2, which occurs at $x = \frac{\pi}{2}$.

Q: What is the minimum value of the graph of $y = -2 \sin(x) - 1$?

A: The minimum value of the graph of $y = -2 \sin(x) - 1$ is -3, which occurs at $x = \frac{3\pi}{2}$.

Q: How does the graph of $y = -2 \sin(x) - 1$ differ from its parent function $y = \sin(x)$?

A: The graph of $y = -2 \sin(x) - 1$ differs from its parent function $y = \sin(x)$ in several key ways. The graph is a vertical stretch of the parent function by a factor of 2, a reflection across the x-axis, and a vertical shift down by 1 unit.

Q: What are some key points used for graphing the function $y = -2 \sin(x) - 1$?

A: Some key points used for graphing the function $y = -2 \sin(x) - 1$ include:

• Period: The period of the graph is still $2\pi$, which means it repeats every $2\pi$ units.
• Amplitude: The amplitude of the graph is 2, which means it oscillates between -2 and 2.
• Maximum Value: The maximum value of the graph is -2, which occurs at $x = \frac{\pi}{2}$.
• Minimum Value: The minimum value of the graph is -3, which occurs at $x = \frac{3\pi}{2}$.

Q: How can I graph the function $y = -2 \sin(x) - 1$?

A: To graph the function $y = -2 \sin(x) - 1$, you can use a graphing calculator or a computer program such as Desmos or GeoGebra. You can also use a piecewise function to graph the function.

Q: What are some real-world applications of the graph of $y = -2 \sin(x) - 1$?

A: The graph of $y = -2 \sin(x) - 1$ has several real-world applications, including:

• Physics: The graph can be used to model the motion of a pendulum or a spring.
• Engineering: The graph can be used to model the behavior of electrical circuits or mechanical systems.
• Biology: The graph can be used to model the growth of populations or the behavior of biological systems.

Q: How can I use the graph of $y = -2 \sin(x) - 1$ to solve problems?

A: You can use the graph of $y = -2 \sin(x) - 1$ to solve problems by identifying the key points of the graph, such as the period, amplitude, maximum value, and minimum value. You can also use the graph to model real-world systems and make predictions about their behavior.