The Graph Of A Sine Function Has An Amplitude Of 2, A Midline Of $y = 10$, And A Period Of 4. There Is No Phase Shift. The Graph Is Reflected Over The $x$-axis.What Is The Equation Of The Function?A. $f(x) = -2 \sin

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Introduction

In mathematics, the graph of a sine function is a fundamental concept that is used to model various real-world phenomena, such as sound waves, light waves, and population growth. The graph of a sine function is characterized by its amplitude, midline, period, and phase shift. In this article, we will explore the equation of a sine function with a given amplitude, midline, period, and phase shift, and then reflect it over the x-axis.

Understanding the Given Information

The given information about the graph of the sine function is as follows:

  • Amplitude: The amplitude of the graph is 2, which means that the maximum value of the function is 2 and the minimum value is -2.
  • Midline: The midline of the graph is y = 10, which means that the graph is shifted 10 units upwards.
  • Period: The period of the graph is 4, which means that the graph repeats itself every 4 units.
  • Phase Shift: There is no phase shift, which means that the graph is not shifted horizontally.
  • Reflection: The graph is reflected over the x-axis, which means that the graph is flipped upside down.

The General Equation of a Sine Function

The general equation of a sine function is given by:

f(x) = A sin(B(x - C)) + D

where:

  • A is the amplitude of the function
  • B is the frequency of the function, which is related to the period by the equation B = 2Ï€ / P
  • C is the phase shift of the function
  • D is the midline of the function

Finding the Equation of the Sine Function

To find the equation of the sine function, we need to substitute the given values into the general equation.

  • Amplitude: A = 2
  • Midline: D = 10
  • Period: P = 4, so B = 2Ï€ / 4 = Ï€/2
  • Phase Shift: C = 0, since there is no phase shift
  • Reflection: Since the graph is reflected over the x-axis, we need to multiply the function by -1.

Substituting these values into the general equation, we get:

f(x) = -2 sin(Ï€/2(x - 0)) + 10

Simplifying the equation, we get:

f(x) = -2 sin(Ï€x/2) + 10

Conclusion

In this article, we have explored the equation of a sine function with a given amplitude, midline, period, and phase shift, and then reflected it over the x-axis. We have used the general equation of a sine function and substituted the given values to find the equation of the sine function. The final equation is f(x) = -2 sin(Ï€x/2) + 10.

Discussion

  • Reflection: When a graph is reflected over the x-axis, the function is multiplied by -1. This is because the reflection is a mirror image of the original graph.
  • Amplitude: The amplitude of a sine function is the maximum value of the function. In this case, the amplitude is 2, which means that the maximum value of the function is 2 and the minimum value is -2.
  • Midline: The midline of a sine function is the average value of the function. In this case, the midline is y = 10, which means that the graph is shifted 10 units upwards.
  • Period: The period of a sine function is the length of one complete cycle of the function. In this case, the period is 4, which means that the graph repeats itself every 4 units.

References

  • Mathematics: The graph of a sine function is a fundamental concept in mathematics that is used to model various real-world phenomena.
  • Trigonometry: The graph of a sine function is related to the trigonometric functions sine, cosine, and tangent.
  • Calculus: The graph of a sine function is used in calculus to model various real-world phenomena, such as population growth and sound waves.
    The Graph of a Sine Function: Q&A =====================================

Introduction

In our previous article, we explored the equation of a sine function with a given amplitude, midline, period, and phase shift, and then reflected it over the x-axis. In this article, we will answer some frequently asked questions about the graph of a sine function.

Q&A

Q: What is the amplitude of a sine function?

A: The amplitude of a sine function is the maximum value of the function. In the case of the graph we explored earlier, the amplitude is 2, which means that the maximum value of the function is 2 and the minimum value is -2.

Q: What is the midline of a sine function?

A: The midline of a sine function is the average value of the function. In the case of the graph we explored earlier, the midline is y = 10, which means that the graph is shifted 10 units upwards.

Q: What is the period of a sine function?

A: The period of a sine function is the length of one complete cycle of the function. In the case of the graph we explored earlier, the period is 4, which means that the graph repeats itself every 4 units.

Q: What is the phase shift of a sine function?

A: The phase shift of a sine function is the horizontal shift of the graph. In the case of the graph we explored earlier, there is no phase shift, which means that the graph is not shifted horizontally.

Q: What happens when a graph is reflected over the x-axis?

A: When a graph is reflected over the x-axis, the function is multiplied by -1. This is because the reflection is a mirror image of the original graph.

Q: How do you find the equation of a sine function?

A: To find the equation of a sine function, you need to substitute the given values into the general equation of a sine function. The general equation of a sine function is given by:

f(x) = A sin(B(x - C)) + D

where:

  • A is the amplitude of the function
  • B is the frequency of the function, which is related to the period by the equation B = 2Ï€ / P
  • C is the phase shift of the function
  • D is the midline of the function

Q: What is the significance of the graph of a sine function in real-world applications?

A: The graph of a sine function is used to model various real-world phenomena, such as sound waves, light waves, and population growth. It is also used in calculus to model various real-world phenomena, such as population growth and sound waves.

Conclusion

In this article, we have answered some frequently asked questions about the graph of a sine function. We have explored the amplitude, midline, period, and phase shift of a sine function, and how to find the equation of a sine function. We have also discussed the significance of the graph of a sine function in real-world applications.

Discussion

  • Real-world applications: The graph of a sine function is used to model various real-world phenomena, such as sound waves, light waves, and population growth.
  • Calculus: The graph of a sine function is used in calculus to model various real-world phenomena, such as population growth and sound waves.
  • Trigonometry: The graph of a sine function is related to the trigonometric functions sine, cosine, and tangent.

References

  • Mathematics: The graph of a sine function is a fundamental concept in mathematics that is used to model various real-world phenomena.
  • Trigonometry: The graph of a sine function is related to the trigonometric functions sine, cosine, and tangent.
  • Calculus: The graph of a sine function is used in calculus to model various real-world phenomena, such as population growth and sound waves.