Amplitude Of Y = -1/3 Sin(x): How To Find It?

by Alex Johnson 46 views

Let's dive into the world of trigonometric functions and specifically focus on understanding the amplitude of the sine function. In this article, we'll break down the function y = -1/3 sin(x) and explore how to determine its amplitude. We'll cover the basic concepts, provide a step-by-step explanation, and clarify why the answer is what it is. Whether you're a student grappling with trigonometry or just curious about the behavior of sine functions, this guide will help you grasp the concept of amplitude with ease.

What is Amplitude?

Before we tackle the specific function, let's define what amplitude means in the context of trigonometric functions. Amplitude is essentially the distance from the center line of the function to its peak or trough. Think of it as the "height" of the wave. For sine and cosine functions, which oscillate above and below the x-axis, the amplitude tells us how far the function strays from this central axis. Understanding amplitude is crucial because it helps us visualize and interpret the behavior of these functions. A larger amplitude means a "taller" wave, while a smaller amplitude indicates a "shorter" wave. This is a fundamental concept in various fields, including physics, engineering, and, of course, mathematics.

The General Form of a Sine Function

The general form of a sine function is given by:

y = A sin(Bx - C) + D

Where:

  • A represents the amplitude. It's the absolute value of the coefficient multiplying the sine function.
  • B affects the period of the function.
  • C represents the phase shift (horizontal shift).
  • D represents the vertical shift.

In our case, the function is y = -1/3 sin(x). Comparing this to the general form, we can see that A is related to -1/3, B is 1, C is 0, and D is 0. We'll focus primarily on A since that's what determines the amplitude.

Identifying the Amplitude in y = -1/3 sin(x)

Now, let’s zoom in on our specific function: y = -1/3 sin(x). As we identified earlier, the coefficient in front of the sine function is -1/3. This value plays a crucial role in determining the amplitude of the function. Remember, the amplitude is the absolute value of this coefficient. This is because the amplitude represents a distance, and distances are always non-negative. So, even though we have a negative sign in front of the fraction, we'll take the absolute value to find the amplitude.

To find the amplitude, we take the absolute value of -1/3:

| -1/3 | = 1/3

Therefore, the amplitude of the function y = -1/3 sin(x) is 1/3. This means the sine wave will oscillate between +1/3 and -1/3 on the y-axis. The negative sign in the original coefficient (-1/3) indicates a reflection across the x-axis, but it doesn't affect the amplitude. The amplitude remains a positive value, representing the maximum displacement from the x-axis.

Why is the Absolute Value Important?

The emphasis on the absolute value is a critical point to understand. The amplitude, by definition, is a measure of distance or displacement. Distance cannot be negative; it’s always a positive quantity (or zero). Think of it this way: the wave moves 1/3 units away from the center line, regardless of whether it’s moving upwards or downwards. The negative sign in front of the 1/3 simply tells us that the wave is flipped upside down compared to a standard y = sin(x) wave. It indicates a reflection across the x-axis. This reflection affects the phase of the sine wave, but it does not change the magnitude of its oscillations. Therefore, we always consider the absolute value to get the true amplitude. Failing to take the absolute value can lead to a misunderstanding of the wave's vertical extent.

Visualizing the Function

To solidify this concept, imagine graphing the function y = -1/3 sin(x). You’ll see a sine wave that oscillates between 1/3 and -1/3. The central line of the wave is the x-axis (y = 0). The highest point the wave reaches is 1/3, and the lowest point is -1/3. The distance from the center line to either of these points is 1/3, which visually confirms our calculated amplitude. Furthermore, the negative sign flips the wave, so it starts by going downwards instead of upwards, which is characteristic of y = sin(x). Despite this flip, the vertical stretch of the wave remains defined by the amplitude.

Comparing with y = sin(x)

Let’s compare our function y = -1/3 sin(x) with the basic sine function, y = sin(x). The function y = sin(x) has an amplitude of 1. This means it oscillates between 1 and -1. Our function, y = -1/3 sin(x), has a smaller amplitude of 1/3, so it's a vertically compressed version of the basic sine wave. It doesn’t reach as high or as low as y = sin(x). The negative sign, as we’ve discussed, simply reflects the wave. By comparing these two functions, we can clearly see the effect of the coefficient on the amplitude and the overall shape of the wave.

Why Options A, B, and D are Incorrect

Now, let’s address why the other options provided in the original question are incorrect:

  • A. -1/3: This is the coefficient in front of the sine function, but it's the negative value. The amplitude is the absolute value, which must be positive.
  • B. Ď€/3: This value doesn't relate to the amplitude. It might be a distractor related to the period or phase shift, but it's not relevant in this context.
  • D. 3: This is the reciprocal of the absolute value of the coefficient, but it doesn't represent the amplitude. It's a common mistake to confuse the coefficient with its reciprocal.

Common Mistakes to Avoid

When determining the amplitude, there are a few common mistakes students often make. One of the biggest errors is forgetting to take the absolute value. Always remember that amplitude is a distance and must be a positive value. Another mistake is confusing the amplitude with other parameters of the sine function, such as the period or phase shift. Keep in mind that the amplitude is solely determined by the coefficient in front of the sine function. Finally, some students might mistakenly focus on the sign of the coefficient. While the sign indicates a reflection, it doesn’t affect the amplitude’s magnitude.

Real-World Applications of Amplitude

Understanding amplitude isn't just an abstract mathematical concept; it has numerous real-world applications. In physics, amplitude is crucial in describing wave phenomena, such as sound waves and light waves. The amplitude of a sound wave corresponds to its loudness, while the amplitude of a light wave corresponds to its brightness. In electrical engineering, amplitude is used to describe the strength of an electrical signal. Understanding amplitude helps engineers design and analyze circuits and communication systems. In music, amplitude relates to the volume of a sound. These are just a few examples, highlighting the practical significance of understanding amplitude in various fields.

Conclusion: The Amplitude of y = -1/3 sin(x) is 1/3

In conclusion, the amplitude of the function y = -1/3 sin(x) is 1/3. We arrived at this answer by understanding the definition of amplitude as the absolute value of the coefficient in front of the sine function. We also discussed why the other options are incorrect and highlighted the importance of taking the absolute value. By visualizing the function and comparing it with the basic sine wave, we reinforced the concept. Understanding amplitude is crucial not just for mathematics but also for various real-world applications. We hope this comprehensive explanation has clarified the concept of amplitude and equipped you with the tools to determine it for any sine function you encounter.

For further exploration of trigonometric functions and their properties, you might find resources on websites like Khan Academy's Trigonometry Section helpful.