Horizontal Shift Of Sine Function Y = Sin(x - 3π/2) Explained

Understanding transformations of trigonometric functions, particularly sine functions, is crucial in mathematics. The question at hand involves analyzing the graph of y = sin(x - 3π/2) and determining how it relates to the graph of the standard sine function, y = sin(x). This requires understanding the concept of phase shift, which is a horizontal translation of a trigonometric function. This article will delve into the specifics of this transformation, providing a comprehensive explanation to clarify the shift in direction and distance. We will dissect the equation, explain the underlying principles of horizontal shifts, and offer visual analogies to solidify understanding. This will ensure that readers grasp not just the answer, but also the mathematical reasoning behind it. This foundational knowledge is essential for tackling more complex trigonometric problems and applications. By the end of this article, you will have a clear understanding of how horizontal shifts work and be able to apply this knowledge to various trigonometric functions.

Decoding the Phase Shift: Horizontal Transformations

When dealing with trigonometric functions, horizontal shifts, also known as phase shifts, can seem tricky at first. However, by understanding the basic principles, we can easily decipher these transformations. The general form of a sine function with a phase shift is y = sin(x - c), where 'c' represents the amount of the horizontal shift. The key point to remember is that the shift is opposite to the sign of 'c'. For instance, if 'c' is positive, the graph shifts to the right, and if 'c' is negative, the graph shifts to the left. This counter-intuitive nature often causes confusion, but it stems from the fact that we are looking at how the input to the sine function is changing. To truly grasp this, it's helpful to visualize the sine wave and how different values of 'x' affect the output. Think about it this way: if you subtract a value from 'x' inside the sine function, you need a larger 'x' value to achieve the same output as the original sine function. This larger 'x' value signifies a shift to the right. Conversely, adding a value to 'x' means you need a smaller 'x' to get the same output, indicating a shift to the left. This fundamental concept underlies all horizontal transformations of trigonometric functions, and mastering it is essential for understanding more complex transformations.

Analyzing y = sin(x - 3π/2): Direction and Magnitude of the Shift

Now, let's apply this understanding to the specific equation given: y = sin(x - 3π/2). Here, we can clearly see that 'c' is equal to 3π/2. Since 3π/2 is a positive value, the graph of y = sin(x - 3π/2) is shifted to the right compared to the graph of y = sin(x). The magnitude of the shift is precisely 3π/2 units. This means that every point on the graph of y = sin(x) is moved 3π/2 units to the right to obtain the graph of y = sin(x - 3π/2). To visualize this, imagine the standard sine wave, which starts at the origin and oscillates between -1 and 1. Now, picture picking up that entire wave and sliding it 3π/2 units to the right along the x-axis. The resulting wave represents the graph of y = sin(x - 3π/2). It's important to note that the shape of the wave remains the same; only its position on the x-axis changes. This rightward shift of 3π/2 units is the key transformation to understand in this context. By recognizing this, you can accurately sketch the graph of y = sin(x - 3π/2) and understand its relationship to the standard sine function.

Visualizing the Shift: Graphing the Functions

To further solidify your understanding, let's visualize the graphs of both y = sin(x) and y = sin(x - 3π/2). The graph of y = sin(x) is the standard sine wave, oscillating between -1 and 1, crossing the x-axis at multiples of π (0, π, 2π, etc.). Its period is 2π, meaning it completes one full cycle over an interval of 2π. Now, consider the graph of y = sin(x - 3π/2). As we've established, this is the same sine wave shifted 3π/2 units to the right. This shift has some interesting consequences. For example, the point where y = sin(x) crosses the x-axis at x = 0, the graph of y = sin(x - 3π/2) crosses the x-axis at x = 3π/2. Similarly, the peak of the sine wave, which occurs at x = π/2 for y = sin(x), occurs at x = π/2 + 3π/2 = 2π for y = sin(x - 3π/2). By plotting key points like these, you can clearly see the horizontal shift in action. Furthermore, it's worth noting that the graph of y = sin(x - 3π/2) is actually equivalent to the graph of y = -cos(x). This trigonometric identity provides another perspective on the transformation, highlighting the connection between sine and cosine functions. Visualizing these graphs and understanding their relationships is a powerful tool for mastering trigonometric transformations.

Why the Shift is to the Right: A Deeper Dive

The question naturally arises: why does subtracting 3π/2 inside the sine function result in a shift to the right? The key to understanding this lies in thinking about the input required to achieve the same output. Consider the point where y = sin(x) reaches its peak value of 1. This occurs at x = π/2. Now, to achieve the same peak value of 1 for y = sin(x - 3π/2), the expression inside the sine function, (x - 3π/2), must also equal π/2. So, we have the equation: x - 3π/2 = π/2. Solving for x, we get x = 2π. This means that the graph of y = sin(x - 3π/2) reaches its peak at x = 2π, whereas the graph of y = sin(x) reaches its peak at x = π/2. The difference between these x-values is 2π - π/2 = 3π/2, which confirms the rightward shift. This example illustrates the general principle: subtracting a value inside the function requires a larger input value to achieve the same output, hence the shift to the right. Conversely, adding a value inside the function would require a smaller input value, resulting in a shift to the left. This understanding is crucial for predicting the behavior of trigonometric functions under various transformations.

Conclusion: Summarizing the Horizontal Shift

In conclusion, the graph of y = sin(x - 3π/2) is the graph of y = sin(x) shifted 3π/2 units to the right. This horizontal shift, or phase shift, is a fundamental concept in understanding trigonometric transformations. By recognizing the form y = sin(x - c), where 'c' represents the magnitude and direction of the shift, we can accurately predict how the graph will be transformed. Remember that a positive 'c' corresponds to a shift to the right, while a negative 'c' corresponds to a shift to the left. Visualizing the graphs and understanding the underlying principles of how the input affects the output are crucial for mastering these transformations. This knowledge will not only help you solve problems involving phase shifts but also build a strong foundation for more advanced topics in trigonometry and calculus. By diligently practicing and applying these concepts, you can confidently navigate the world of trigonometric functions and their transformations.

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