Finding The Range Of 2sin(x): A Simple Guide

Oct 14, 2025 by ADMIN 45 views

Hey everyone! Let's dive into a classic math question: What is the range of the function y = 2sin(x)? Understanding the range of trigonometric functions like this is super important in math, and it's not as scary as it might seem at first glance. In this guide, we'll break it down step by step, so you'll be a pro in no time. We will also touch on the other answer options. We will provide the correct option and also provide the reason why other answers are wrong.

Understanding the Basics: What is Range?

First things first, let's make sure we're all on the same page about what range means. The range of a function is the set of all possible output values (y-values) that the function can produce. Think of it like this: you plug in a bunch of x-values, and the function spits out a bunch of y-values. The range is simply the collection of all those y-values. For example, if we are given a simple function f(x) = x^2, we know that every result of this function will be positive or 0. That means the range is all positive numbers, as well as 0. The same is true for trigonometric functions. The difference is that we have some special rules for the range of these functions, so the y-values can be different. Now, let's apply this concept to our function, y = 2sin(x).

To truly grasp the range of y = 2sin(x), we first need to understand the behavior of the basic sine function, sin(x). The sine function is a periodic function, meaning it repeats its values over and over again. If we consider the unit circle, the values of sin(x) correspond to the y-coordinates of points on the circle. Because of this, the sine function oscillates between -1 and 1. It never goes higher than 1 or lower than -1. So, the range of sin(x) is [-1, 1]. This is a critical piece of information for finding the range of our function.

Now, consider the graph of y = sin(x). It's a wave that goes up to 1 and down to -1. When we introduce the constant '2' as in y = 2sin(x), we're essentially stretching the graph vertically. The graph will be vertically stretched by a factor of 2. The original peak and trough will be affected, but it does not change any other aspect of the function. This means that the y-values are multiplied by 2. So the maximum value, which was 1, becomes 2, and the minimum value, which was -1, becomes -2. This vertical stretch is what changes the range.

So, why is understanding the range of sin(x) important? Because it gives us a foundation. By knowing the possible outputs of sin(x), we can determine the range of y = 2sin(x). The '2' in the equation y = 2sin(x) simply stretches the sine wave vertically, which changes its amplitude. Amplitude is the distance from the center line (the x-axis in this case) to the peak or trough of the wave. For sin(x), the amplitude is 1. For 2sin(x), the amplitude is 2. This understanding is crucial for solving a variety of trigonometric problems, and knowing the range is the first step in many of these. It helps us to understand the limits of the function's output, which is essential for understanding its behavior.

Analyzing the Options

Let's go through the answer choices to see which one is correct and why the others are wrong. This is always a great way to solidify your understanding, so you will be good at it! We are going to focus on the question: What is the range of the function y = 2sin(x)?

A. [-2, 2]: This is the correct answer! As we discussed, the sine function oscillates between -1 and 1. Multiplying the sine function by 2 stretches the graph vertically, so the range becomes all values between -2 and 2, inclusive. The maximum value of the function is 2 (when sin(x) = 1), and the minimum value is -2 (when sin(x) = -1). So the range is effectively [-2, 2]. You can also think of this like the following: -1 <= sin(x) <= 1, we multiply by 2, resulting in -2 <= 2sin(x) <= 2. The conclusion is, the range of 2sin(x) is indeed [-2, 2].
B. All real numbers: This option is incorrect. This is because the sine function, and thus any scaled version of it, is limited by its oscillatory nature. The sine wave never goes to infinity or negative infinity. Because of this, the outputs of 2sin(x) will always be between -2 and 2. The range is not all real numbers. The function does not output all possible numbers, only a limited set.
C. 2π: This option is incorrect because 2π represents the period of the sine function, not its range. The period is the length of one complete cycle of the wave. The function repeats itself every 2π radians or 360 degrees. The range refers to the set of all the possible y-values that the function produces, and it does not have anything to do with the length of one cycle of the wave. The range of 2sin(x) is not a single value, like 2π.
D. π: This option is incorrect for the same reason as option C. π represents half of the period of the sine function. It is not related to the range. The same logic applies. The range is the y-values that the function can produce. It does not have anything to do with the period.

Conclusion: Mastering the Range of 2sin(x)

So there you have it, guys! The range of the function y = 2sin(x) is [-2, 2]. We've covered the basics of range, the behavior of the sine function, and how multiplying the sine function by a constant affects its range. Understanding this concept is fundamental for working with trigonometric functions. Keep practicing, and you'll become a pro in no time. Don't be afraid to break down complex problems into smaller, more manageable steps. When dealing with trigonometric functions, remember the basic properties of sine, cosine, and tangent. This knowledge will help you solve a variety of problems. The range is just one piece of the puzzle. When you encounter similar problems in the future, you'll be able to confidently identify the range, and your understanding of trigonometric functions will be further reinforced. Keep practicing, and you'll be mastering these concepts in no time. Good luck and keep learning!