Transforming the Sine Function

Oct 13, 2025 by windailys.com 59 views

Introduction

In mathematics, the range of a function is the set of all possible output values it can produce for the given input values. When dealing with trigonometric functions, such as the sine function, understanding the range is crucial in determining the behavior of the function. In this article, we will explore the range of the function $$y = -3 \sin(x) - 4$$ and determine the correct answer among the given options.

The Sine Function

The sine function is a periodic function that oscillates between -1 and 1. Its general form is $$y = \sin(x)$$, where $$x$$ is the input value and $$y$$ is the output value. The sine function has a range of $$[-1, 1]$$, meaning that the output value $$y$$ can take any value between -1 and 1.

Transforming the Sine Function

The given function $$y = -3 \sin(x) - 4$$ is a transformation of the basic sine function. The transformation involves two main components:

Vertical Stretching: The coefficient $$-3$$ in front of the sine function stretches the graph vertically by a factor of 3. This means that the amplitude of the function increases by a factor of 3.
Vertical Translation: The constant term $$-4$$ shifts the graph vertically downward by 4 units. This means that the function is translated by 4 units in the downward direction.

Understanding the Range

To determine the range of the function $$y = -3 \sin(x) - 4$$, we need to consider the effects of the vertical stretching and translation on the basic sine function.

Vertical Stretching: The vertical stretching by a factor of 3 increases the amplitude of the function. Since the original range of the sine function is $$[-1, 1]$$, the new range after stretching will be $$[-3, 3]$$.
Vertical Translation: The vertical translation by 4 units downward shifts the range of the function. Since the function is translated downward, the new range will be $$[-3 - 4, 3 - 4]$$, which simplifies to $$[-7, -1]$$.

Conclusion

Based on the analysis of the function $$y = -3 \sin(x) - 4$$, we can conclude that the range of the function is all real numbers $$-7 \leq y \leq -1$$. This means that the output value $$y$$ can take any value between -7 and -1.

Answer

The correct answer is:

B. All real numbers $$-7 \leq y \leq -1$$

Final Thoughts

Introduction

In our previous article, we explored the range of the function $$y = -3 \sin(x) - 4$$ and determined that the correct answer is all real numbers $$-7 \leq y \leq -1$$. In this article, we will address some common questions and concerns related to the range of a sine function.

Q: What is the range of a basic sine function?

A: The range of a basic sine function is $$[-1, 1]$$, meaning that the output value $$y$$ can take any value between -1 and 1.

Q: How does vertical stretching affect the range of a sine function?

A: Vertical stretching by a factor of $$k$$ increases the amplitude of the function, resulting in a new range of $$[-k, k]$$.

Q: How does vertical translation affect the range of a sine function?

A: Vertical translation by $$c$$ units shifts the range of the function, resulting in a new range of $$[-k - c, k - c]$$.

Q: What is the range of the function $$y = 2 \sin(x) + 3$$?

A: To determine the range of this function, we need to consider the effects of vertical stretching and translation.

Vertical Stretching: The coefficient $$2$$ in front of the sine function stretches the graph vertically by a factor of 2. This means that the amplitude of the function increases by a factor of 2.
Vertical Translation: The constant term $$3$$ shifts the graph vertically upward by 3 units. This means that the function is translated by 3 units in the upward direction.

Using the formulas for vertical stretching and translation, we can determine the new range of the function:

Vertical Stretching: The new range after stretching will be $$[-2, 2]$$.
Vertical Translation: The new range after translation will be $$[-2 + 3, 2 + 3]$$, which simplifies to $$[1, 5]$$.

Therefore, the range of the function $$y = 2 \sin(x) + 3$$ is all real numbers $$1 \leq y \leq 5$$.

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

A: To determine the range of this function, we need to consider the effects of vertical stretching and translation.

Vertical Stretching: The coefficient $$-2$$ in front of the sine function stretches the graph vertically by a factor of 2. This means that the amplitude of the function increases by a factor of 2.
Vertical Translation: The constant term $$-2$$ shifts the graph vertically downward by 2 units. This means that the function is translated by 2 units in the downward direction.

Using the formulas for vertical stretching and translation, we can determine the new range of the function:

Vertical Stretching: The new range after stretching will be $$[-2, 2]$$.
Vertical Translation: The new range after translation will be $$[-2 - 2, 2 - 2]$$, which simplifies to $$[-4, 0]$$.

Therefore, the range of the function $$y = -2 \sin(x) - 2$$ is all real numbers $$-4 \leq y \leq 0$$.

Conclusion

Understanding the range of a sine function is crucial in mathematics, especially when dealing with trigonometric functions. By analyzing the effects of transformations on the basic sine function, we can determine the range of the function and make informed decisions about its behavior.