What Is The Range Of Y = -3sin(x)?

In the realm of trigonometry, understanding the range of functions is crucial for grasping their behavior and applications. The range of a function defines the set of all possible output values. This article delves into determining the range of the function y = -3sin(x), a fundamental concept for anyone studying mathematics, physics, or engineering. Specifically, we aim to provide a comprehensive explanation that helps you understand how the coefficient of the sine function affects its range. By the end of this exploration, you'll not only know the answer but also the underlying principles that govern trigonometric functions.

The Foundation: Understanding the Sine Function

Before we tackle y = -3sin(x), it's essential to solidify our understanding of the basic sine function, y = sin(x). The sine function is a cornerstone of trigonometry, mapping angles to values between -1 and 1. To truly grasp the range of y = sin(x), let's dissect its behavior across the unit circle. Imagine a point moving counterclockwise around the circle. The y-coordinate of this point represents the sine of the angle formed with the positive x-axis. As the point traverses the circle, the y-coordinate oscillates between -1 and 1. At 0 degrees (or 0 radians), the y-coordinate is 0, so sin(0) = 0. As we move to 90 degrees (π/2 radians), the y-coordinate reaches its maximum value of 1, thus sin(π/2) = 1. Continuing to 180 degrees (π radians), the y-coordinate returns to 0, making sin(π) = 0. At 270 degrees (3π/2 radians), the y-coordinate hits its minimum value of -1, resulting in sin(3π/2) = -1. Finally, completing the circle at 360 degrees (2π radians), the y-coordinate is back at 0, so sin(2π) = 0. This cyclical behavior defines the sine wave, a smooth, continuous oscillation that repeats every 2π radians. The maximum value of sin(x) is 1, occurring at angles like π/2, 5π/2, and so on. Conversely, the minimum value is -1, seen at angles such as 3π/2, 7π/2, and so forth. Because the sine function oscillates between -1 and 1, its range is defined as -1 ≤ y ≤ 1. This range is a fundamental property of the sine function and serves as a building block for understanding more complex trigonometric functions. Grasping this concept is key to predicting how modifications to the sine function, such as the introduction of coefficients or phase shifts, will affect its range. Understanding the range of the sine function is not merely an academic exercise; it has practical implications across various fields. For instance, in physics, the sine function is used to model simple harmonic motion, such as the oscillation of a pendulum or the vibration of a string. The range of the sine function directly corresponds to the amplitude of these oscillations, representing the maximum displacement from the equilibrium position. In electrical engineering, alternating current (AC) waveforms are often described using sine functions, where the range determines the peak voltage or current. Therefore, a solid understanding of the sine function's range is essential for analyzing and designing electrical circuits and systems. By understanding this foundation, we can then explore how transformations affect this basic range, setting the stage for analyzing y = -3sin(x).

The Impact of the Coefficient: Stretching and Reflecting

Now that we have a firm grasp of the basic sine function, let's explore how multiplying it by a coefficient changes its behavior, focusing specifically on y = -3sin(x). The coefficient in front of the sine function plays a crucial role in determining the function's amplitude and, consequently, its range. In this case, the coefficient is -3, which introduces two key transformations to the basic sine wave: a vertical stretch and a reflection across the x-axis. First, consider the absolute value of the coefficient, which is |–3| = 3. This value represents the amplitude of the function, defining the maximum displacement from the x-axis. Multiplying sin(x) by 3 stretches the function vertically, causing it to oscillate between -3 and 3 instead of -1 and 1. This means that the function's highest point is now 3 and its lowest point is -3. The positive and negative signs indicate the direction and magnitude of displacement from the x-axis. The second transformation is the reflection across the x-axis, which arises from the negative sign of the coefficient. The negative sign in -3sin(x) flips the sine wave vertically. Where sin(x) would have been positive, -3sin(x) is negative, and vice versa. This reflection does not change the amplitude but inverts the function's behavior, essentially mirroring it along the x-axis. The combined effect of the vertical stretch and reflection profoundly impacts the range of the function. The stretching amplifies the oscillation, while the reflection flips the wave. To visualize this, imagine the original sine wave oscillating between -1 and 1. Multiplying by 3 extends these bounds to -3 and 3, and the negative sign flips the wave so that the peaks become troughs and vice versa. Consequently, the range of y = -3sin(x) is determined by these new boundaries. The function will now oscillate between -3 and 3, inclusive. Thus, the range is –3 ≤ y ≤ 3. This range represents all possible output values of the function, and no value will fall outside this interval. In summary, the coefficient of -3 in y = -3sin(x) not only stretches the sine function vertically by a factor of 3 but also reflects it across the x-axis. Understanding the individual and combined effects of these transformations is essential for determining the range of the function. This principle extends to other trigonometric functions as well, where coefficients can significantly alter their ranges and behaviors. By mastering this concept, you'll gain a deeper insight into the nature of trigonometric functions and their applications in various mathematical and scientific contexts. This understanding also lays the groundwork for analyzing more complex trigonometric transformations, such as phase shifts and horizontal stretches, which further modify the behavior of these functions. The coefficient's impact is not limited to just stretching and reflecting; it also influences the function's energy and the scale of the phenomena it models. For example, in physics, a larger amplitude in a wave function signifies a greater amount of energy carried by the wave. In electrical signals, a higher amplitude translates to a stronger signal. Therefore, the coefficient provides vital information about the function's characteristics and its physical implications. Grasping the nuances of how coefficients affect trigonometric functions empowers you to interpret and apply these functions in a wider range of scenarios.

Determining the Range of y = -3sin(x): A Step-by-Step Approach

To definitively determine the range of y = -3sin(x), let's follow a step-by-step approach. This method not only provides the answer but also reinforces the underlying principles. The sine function, y = sin(x), oscillates between -1 and 1, inclusive. This fundamental property is our starting point. The range of y = sin(x) is -1 ≤ sin(x) ≤ 1. Next, consider the coefficient in front of the sine function, which is -3. This coefficient will modify the range by stretching and possibly reflecting the sine wave. Multiply all parts of the inequality by -3. Remember, when multiplying an inequality by a negative number, the direction of the inequality signs must be reversed. Starting with -1 ≤ sin(x) ≤ 1, multiply each part by -3: (-1) * (-3) ≥ -3sin(x) ≥ (1) * (-3). This simplifies to 3 ≥ -3sin(x) ≥ -3. To make the inequality more intuitive, rewrite it with the smaller value on the left: -3 ≤ -3sin(x) ≤ 3. This inequality directly gives us the range of y = -3sin(x). The function y = -3sin(x) oscillates between -3 and 3, inclusive. Therefore, the range of y = -3sin(x) is -3 ≤ y ≤ 3. This means that the function's output values will never be less than -3 or greater than 3. Visualizing the graph of y = -3sin(x) confirms this result. The sine wave is stretched vertically by a factor of 3 and reflected across the x-axis, causing it to oscillate between the lines y = -3 and y = 3. The graph's peaks reach a maximum value of 3, and its troughs reach a minimum value of -3. This visual representation solidifies the understanding of the function's range. It's essential to recognize that the range is not merely a set of isolated values but a continuous interval. The function takes on every value between -3 and 3, making the range a smooth, uninterrupted spectrum. This continuity is a characteristic feature of trigonometric functions and reflects the smooth, cyclical nature of sine and cosine waves. By breaking down the process into these steps, we not only find the range but also gain a deeper understanding of how coefficients affect trigonometric functions. This method can be applied to other similar functions, allowing you to confidently determine their ranges. For example, consider the function y = 2sin(x). Following the same steps, you would multiply the inequality -1 ≤ sin(x) ≤ 1 by 2, resulting in -2 ≤ 2sin(x) ≤ 2. Thus, the range of y = 2sin(x) is -2 ≤ y ≤ 2. Similarly, for y = -0.5sin(x), multiplying by -0.5 and reversing the inequality signs yields 0.5 ≥ -0.5sin(x) ≥ -0.5, which simplifies to -0.5 ≤ -0.5sin(x) ≤ 0.5. Therefore, the range of y = -0.5sin(x) is -0.5 ≤ y ≤ 0.5. These examples demonstrate the versatility of this step-by-step approach in finding the range of various sine functions with different coefficients. The method highlights the impact of the coefficient on the amplitude and direction of the sine wave's oscillation. Mastery of this technique is invaluable for anyone working with trigonometric functions, whether in mathematics, physics, engineering, or other fields where periodic phenomena are studied.

Conclusion: The Range of y = -3sin(x) and Its Implications

In conclusion, we've thoroughly explored the range of the function y = -3sin(x). By understanding the basic properties of the sine function and the impact of the coefficient, we've determined that the range of y = -3sin(x) is -3 ≤ y ≤ 3. This range signifies the complete set of possible output values for the function, spanning from -3 to 3, inclusive. The coefficient of -3 plays a pivotal role in shaping this range. The absolute value of the coefficient, |–3| = 3, dictates the amplitude of the function, defining the maximum displacement from the x-axis. The sine wave is stretched vertically by a factor of 3, causing it to oscillate between -3 and 3 instead of -1 and 1. The negative sign of the coefficient introduces a reflection across the x-axis, inverting the sine wave's behavior. This reflection, combined with the vertical stretch, yields the final range. This principle is not confined to y = -3sin(x); it extends to other trigonometric functions as well. Coefficients, phase shifts, and other transformations can significantly alter the range and behavior of these functions. Mastering these concepts is crucial for anyone working with trigonometric functions in various fields, from mathematics and physics to engineering and computer graphics. In physics, understanding the range of sinusoidal functions is essential for analyzing wave phenomena, such as sound waves and electromagnetic waves. The range corresponds to the amplitude of the wave, which determines its energy and intensity. In electrical engineering, alternating current (AC) circuits are often modeled using sine functions, and the range represents the peak voltage or current. Similarly, in computer graphics, sine and cosine functions are used to create smooth animations and movements, where the range influences the extent of the motion. By grasping the underlying principles of trigonometric functions and their transformations, you can confidently apply these tools in a wide range of contexts. The range, in particular, provides vital information about the function's behavior and its physical implications. As you continue your mathematical journey, remember that a deep understanding of fundamental concepts like the range of trigonometric functions will serve as a solid foundation for tackling more complex problems. By mastering these basics, you'll unlock a deeper appreciation for the beauty and power of mathematics and its applications in the world around us. The ability to analyze and interpret trigonometric functions is a valuable skill that will empower you to solve real-world problems and explore new frontiers in science and technology. The journey of mathematical discovery is a continuous one, and each step you take, no matter how small, brings you closer to a more profound understanding of the universe.