Finding The Range Of Y = -5sin(x) A Comprehensive Guide

In the realm of trigonometry, understanding the range of sinusoidal functions is crucial. When we delve into the range of trigonometric functions, particularly those involving sine and cosine, we uncover fundamental properties that govern their behavior. The range, in simple terms, defines the set of all possible output values a function can produce. For a sinusoidal function like y = A sin(Bx + C) + D, the range is influenced by several key parameters, including the amplitude (A), vertical shift (D), and any reflections or stretches. In this article, we will explore in detail the range of the function y = -5sin(x), dissecting how the coefficient -5 affects the sine function's output values. We will clarify why the correct range is -5 ≤ y ≤ 5, and we'll also eliminate other options through clear, step-by-step explanations. Our focus will be on providing a comprehensive and intuitive understanding, suitable for students and enthusiasts alike who aim to master the basics of trigonometric functions. Understanding these concepts not only aids in solving mathematical problems but also provides a solid foundation for more advanced topics in calculus and physics, where sinusoidal functions are frequently encountered.

Before we dive into the specifics of y = -5sin(x), let's establish a firm grasp of the core concepts related to sinusoidal functions. At its heart, the sine function, denoted as sin(x), oscillates between -1 and 1. This is a fundamental characteristic stemming from the unit circle definition of sine, where sin(x) represents the y-coordinate of a point on the unit circle corresponding to an angle x. Thus, for any real number x, the value of sin(x) will always fall within the closed interval [-1, 1]. This intrinsic property is crucial for understanding how transformations, such as vertical stretches, compressions, and reflections, affect the range of the function. When we introduce a coefficient, such as the -5 in our function y = -5sin(x), we essentially multiply the output of the sin(x) function by this coefficient. This multiplication vertically stretches the sine wave and, if the coefficient is negative, reflects it across the x-axis. Consequently, the range of the transformed function will change proportionally. Additionally, it's vital to remember that these transformations do not alter the period of the basic sine function, which remains 2π, unless there's a horizontal stretch or compression. In essence, understanding these fundamental concepts allows us to predict and interpret the behavior of sinusoidal functions under various transformations, making it easier to determine their range and other key characteristics.

Now, let's focus specifically on the function y = -5sin(x) and dissect its range. As we established earlier, the sine function, sin(x), inherently oscillates between -1 and 1. This means that the smallest value sin(x) can take is -1, and the largest value is 1. When we introduce the coefficient -5, we're essentially scaling these boundaries. Multiplying the entire range [-1, 1] by -5 gives us a new range. Let's break it down: when sin(x) = -1, then y = -5 * (-1) = 5. Conversely, when sin(x) = 1, then y = -5 * (1) = -5. This simple calculation reveals that the new range stretches from -5 to 5. The negative sign in -5 not only scales the function but also reflects it across the x-axis. This reflection inverts the typical sine wave, but it does not change the magnitude of the extreme values. Thus, the range of y = -5sin(x) spans all values between -5 and 5, inclusive. This understanding is crucial for accurately graphing the function and interpreting its behavior in various contexts. Furthermore, it highlights the role of the coefficient in vertically stretching and reflecting the sine function, leading to a clear determination of the range.

To understand why -5 ≤ y ≤ 5 is the correct range for y = -5sin(x), let's reiterate the behavior of the sine function and the impact of the coefficient -5. As we know, sin(x) varies between -1 and 1, inclusive. This fundamental property is the cornerstone of our analysis. When we multiply sin(x) by -5, we are essentially transforming the original range. The maximum value of sin(x), which is 1, becomes -5 * 1 = -5. The minimum value of sin(x), which is -1, becomes -5 * (-1) = 5. Therefore, the new maximum value of y is 5, and the new minimum value is -5. This means that y can take any value between -5 and 5, inclusive. It's crucial to note that the sine function is continuous, meaning it smoothly transitions between these extreme values. Thus, y = -5sin(x) covers all real numbers between -5 and 5. This comprehensive coverage is what makes the interval [-5, 5] the correct range. Any value outside this interval cannot be attained by the function y = -5sin(x) because the sine function itself is bounded between -1 and 1. The coefficient -5 merely stretches and reflects the sine wave, but it doesn't alter the fundamental oscillatory nature or introduce values beyond this defined range.

Now, let's systematically eliminate the incorrect options to reinforce our understanding of the range of y = -5sin(x). Option A suggests that the range is all real numbers. This is incorrect because the sine function, regardless of any coefficient, is bounded. It never extends to infinity or negative infinity. The multiplication by -5 stretches and reflects the sine wave, but it doesn't remove these bounds. Option C proposes the range -5/2 ≤ y ≤ 5/2. This is also incorrect because it underestimates the impact of the coefficient -5. As we calculated, multiplying the extreme values of sin(x) by -5 gives us -5 and 5, not -5/2 and 5/2. Option D, which suggests -1 ≤ y ≤ 1, is simply the range of the basic sin(x) function. It neglects the effect of the coefficient -5, which stretches the range by a factor of 5. Option E is incomplete as it doesn't specify the range, but it alludes to real numbers, which, as we discussed, is incorrect for bounded functions like sine. By carefully analyzing each option and comparing it with the derived range -5 ≤ y ≤ 5, we can see that only option B accurately reflects the possible output values of y = -5sin(x). This elimination process further solidifies our grasp on the concept of range and how coefficients affect sinusoidal functions.

A powerful way to further understand the range of y = -5sin(x) is to visualize its graph. When we plot the function, we can clearly see its oscillatory behavior and how the range is constrained between -5 and 5. The graph of y = -5sin(x) is a sine wave that has been vertically stretched by a factor of 5 and reflected across the x-axis due to the negative sign. The reflection means that the wave starts by going downwards instead of upwards, as the basic sin(x) function does. However, the key observation here is that the highest point of the wave reaches y = 5, and the lowest point reaches y = -5. The wave oscillates continuously between these two values, never exceeding or falling below them. This visual representation perfectly matches our mathematical analysis, where we determined that the range is -5 ≤ y ≤ 5. By looking at the graph, it becomes intuitively clear that no y-value outside this interval can be attained by the function. The graph serves as a tangible confirmation of the range, reinforcing our understanding and providing a visual reference for future problems involving sinusoidal functions.

Understanding the range of trigonometric functions like y = -5sin(x) is not just a theoretical exercise; it has practical applications across various fields. Sinusoidal functions are fundamental in modeling periodic phenomena, which occur frequently in nature and engineering. For instance, these functions are used to describe oscillations in mechanical systems, such as the motion of a pendulum or the vibration of a spring. They are also crucial in electrical engineering for analyzing alternating current (AC) circuits, where voltage and current vary sinusoidally over time. In physics, sinusoidal functions describe wave phenomena, including sound waves and electromagnetic waves. The range of these functions is particularly important because it represents the amplitude or maximum displacement of the oscillation or wave. In the case of y = -5sin(x), the range [-5, 5] indicates that the maximum displacement from the equilibrium position is 5 units in either direction. This information can be critical in designing systems that can withstand these oscillations or in interpreting the strength of a signal. Therefore, a solid grasp of the range not only enhances mathematical problem-solving skills but also provides insights into real-world phenomena, making it an essential concept for students and professionals in science, technology, engineering, and mathematics (STEM) fields.

In conclusion, the range of the function y = -5sin(x) is -5 ≤ y ≤ 5. This result stems from the inherent properties of the sine function, which oscillates between -1 and 1, and the impact of the coefficient -5, which stretches the range vertically and reflects the function across the x-axis. We arrived at this conclusion through a detailed analysis, eliminating other options based on logical reasoning and mathematical principles. Visualizing the graph of the function further solidified our understanding, showing how the wave oscillates within the defined range. Moreover, we highlighted the practical significance of understanding the range in various applications, from modeling physical systems to analyzing electrical circuits. This comprehensive exploration not only answers the specific question but also reinforces the fundamental concepts of sinusoidal functions and their behavior under transformations. By mastering these principles, students and professionals can confidently tackle more complex problems involving trigonometric functions and their applications. The ability to accurately determine the range is a critical skill, and our discussion here provides a solid foundation for future learning and application in mathematics, science, and engineering. Therefore, understanding the range of y = -5sin(x) is a cornerstone in the broader study of trigonometric functions and their role in describing the world around us.