Describe The Transformation Of $f(x) = \sin X$ To $g(x) = \sin X - 7$.A. $f(x$\] Is Shifted 7 Units Right To $g(x$\]. B. $f(x$\] Is Shifted 7 Units Down To $g(x$\]. C. $f(x$\] Is Shifted 7

Transformation of Trigonometric Functions: A Case Study of $f(x) = \sin x$ to $g(x) = \sin x - 7$

In mathematics, transformations play a crucial role in understanding the behavior of functions. When a function is transformed, its graph undergoes a change in position, shape, or size. In this article, we will explore the transformation of the function $f(x) = \sin x$ to $g(x) = \sin x - 7$. We will examine the effect of subtracting 7 from the function and determine whether it is shifted right, left, up, or down.

Understanding the Original Function

The original function is $f(x) = \sin x$. This is a basic trigonometric function that represents the sine of an angle. The graph of this function is a periodic curve that oscillates between -1 and 1.

The Transformation

The transformed function is $g(x) = \sin x - 7$. To understand the effect of this transformation, let's analyze the change in the function. When we subtract 7 from the function, we are essentially shifting the graph of the function down by 7 units.

Is the Function Shifted Right or Left?

To determine whether the function is shifted right or left, we need to examine the coefficient of the $x$ term. In the original function, the coefficient of $x$ is 1. In the transformed function, the coefficient of $x$ is still 1. Since the coefficient of $x$ remains the same, the function is not shifted right or left.

Is the Function Shifted Up or Down?

To determine whether the function is shifted up or down, we need to examine the constant term. In the original function, the constant term is 0. In the transformed function, the constant term is -7. Since the constant term is negative, the function is shifted down.

Conclusion

In conclusion, the transformation of $f(x) = \sin x$ to $g(x) = \sin x - 7$ results in a shift of the graph down by 7 units. The function is not shifted right or left, but rather down. This transformation changes the position of the graph, but not its shape or size.

Visualizing the Transformation

To visualize the transformation, let's consider the graph of the original function and the transformed function.

	Original Function	Transformed Function
Graph

As we can see from the graphs, the original function is a basic sine wave, while the transformed function is the same sine wave shifted down by 7 units.

Real-World Applications

The transformation of functions has numerous real-world applications in various fields, including physics, engineering, and economics. For example, in physics, the transformation of functions can be used to model the motion of objects under different forces. In engineering, the transformation of functions can be used to design and optimize systems. In economics, the transformation of functions can be used to model and analyze economic systems.

Conclusion

In our previous article, we explored the transformation of the function $f(x) = \sin x$ to $g(x) = \sin x - 7$. We examined the effect of subtracting 7 from the function and determined that it results in a shift of the graph down by 7 units. In this article, we will answer some frequently asked questions about the transformation of trigonometric functions.

Q: What is the effect of adding a constant to a trigonometric function?

A: When a constant is added to a trigonometric function, the graph of the function is shifted up by the value of the constant. For example, if we add 3 to the function $f(x) = \sin x$, the resulting function is $g(x) = \sin x + 3$, which is shifted up by 3 units.

Q: What is the effect of subtracting a constant from a trigonometric function?

A: When a constant is subtracted from a trigonometric function, the graph of the function is shifted down by the value of the constant. For example, if we subtract 2 from the function $f(x) = \sin x$, the resulting function is $g(x) = \sin x - 2$, which is shifted down by 2 units.

Q: How does the transformation of a trigonometric function affect its amplitude?

A: The transformation of a trigonometric function does not affect its amplitude. The amplitude of a function is the maximum value that the function can attain, and it is determined by the coefficient of the sine or cosine term. When a constant is added or subtracted from a trigonometric function, the amplitude remains the same.

Q: Can the transformation of a trigonometric function change its period?

A: Yes, the transformation of a trigonometric function can change its period. When a constant is added or subtracted from a trigonometric function, the period of the function remains the same if the constant is a multiple of the period. However, if the constant is not a multiple of the period, the period of the function changes.

Q: How does the transformation of a trigonometric function affect its phase shift?

A: The transformation of a trigonometric function can affect its phase shift. When a constant is added or subtracted from a trigonometric function, the phase shift of the function changes. The phase shift is determined by the value of the constant and the period of the function.

Q: Can the transformation of a trigonometric function result in a change in the shape of the graph?

A: No, the transformation of a trigonometric function cannot result in a change in the shape of the graph. The shape of a trigonometric function is determined by the type of function (sine, cosine, etc.) and the coefficient of the sine or cosine term. When a constant is added or subtracted from a trigonometric function, the shape of the graph remains the same.

Conclusion

In conclusion, the transformation of trigonometric functions is an important concept in mathematics that has numerous real-world applications. By understanding how the transformation of a trigonometric function affects its graph, we can better analyze and model complex systems in various fields.