Analyzing Trigonometric Functions: Agreement On F(x) = 2sin(x) Graphs

Hey guys! Let's dive into the awesome world of trigonometry, specifically looking at the function f(x) = 2sin(x). We're going to graph this function on the good ol' Cartesian coordinate system, and the goal is to figure out whether we agree or disagree with some statements about it. This is super helpful for understanding how trigonometric functions work. Remember, the sine function is a fundamental concept, so grasping how it behaves, especially with modifications like the one in f(x) = 2sin(x), is key.

Understanding the Basics of f(x) = 2sin(x)

Alright, before we get into the statements, let's quickly recap what f(x) = 2sin(x) actually is. The basic sine function, sin(x), is a wave that oscillates between -1 and 1. The '2' in front of the sine function, making it 2sin(x), is called the amplitude. This is super important! It tells us how 'tall' the wave is, meaning how far it stretches from the x-axis. In this case, the amplitude is 2, so the wave will oscillate between -2 and 2. The period of the function, which is the length of one complete cycle, remains the same as the basic sine function, which is 2π (or approximately 6.28). There is no horizontal shift happening here, so the graph of f(x) = 2sin(x) starts at the origin (0,0), just like the regular sin(x) function. The graph increases to its maximum value (2) at π/2 (or approximately 1.57), goes back to 0 at π (or approximately 3.14), decreases to its minimum value (-2) at 3π/2 (or approximately 4.71), and returns to 0 at 2π (or approximately 6.28). This is the fundamental behavior of the graph we'll be assessing. Thinking about these characteristics beforehand helps you anticipate the behavior of the function and helps you visualize and understand what is happening on the graph. The amplitude scaling means the wave is stretched vertically, while the period remains unchanged. The shape is still sinusoidal, just with a larger vertical range. This is the foundation for understanding all of the statements about the function.

Knowing these key features allows us to make more accurate predictions. When approaching each of the statements, keep these features in mind. The amplitude is going to be the biggest thing to consider. The amplitude affects the range of the function, or the possible y values, of the sine wave. In this case the range is [-2, 2], from the negative value of the amplitude to the positive value of the amplitude. Be sure to consider the x values as well. The period of the function is the length of one full cycle of the wave. For sin(x) this is 2π, but in the question 2sin(x) the period is unchanged. So, we should still expect the function to return to its starting point after 2π. And finally, recognize the y value when x = 0. This is the starting point of the graph, which we mentioned earlier is (0,0). Now, with the basics in mind, we can start to determine the validity of the provided statements. Let's get to it.

Analyzing Statements: Setuju or Tidak Setuju?

Now, let’s get down to the core of the task: evaluating each statement about the function f(x) = 2sin(x). For each statement, we will determine whether we Setuju (agree) or Tidak Setuju (disagree) based on our understanding of the sine function and the impact of the amplitude.

Statement 1: The amplitude of the function is 1.

This statement talks about the amplitude of the function, and in the case of f(x) = 2sin(x), this should be the easiest one to determine. Remember, the amplitude is the absolute value of the coefficient in front of the sin(x), which in this case is '2'. So, the amplitude represents the maximum displacement from the x-axis. If the amplitude were 1, the graph would oscillate between -1 and 1. However, since it is 2, the graph oscillates between -2 and 2, making the wave twice as tall as the standard sine wave. So, if this statement says the amplitude is 1, we definitely Tidak Setuju. The graph's range is defined by the amplitude. The amplitude is the distance from the center line to the peak or the trough of the sine wave. Since the sine wave oscillates from -2 to 2, it is obvious the amplitude is 2 and not 1. Therefore, we should not agree.

Statement 2: The maximum value of the function is 2.

This is where it gets a little more interesting, and we'll need to apply the concepts we discussed earlier. The maximum value of a sine function directly relates to the amplitude. Since the amplitude is 2, the sine wave extends up to a maximum value of 2. This means the highest point on the graph is at y = 2. The sine function oscillates between a minimum value and a maximum value, both being equidistant from the x-axis. Because of this relationship, this statement is most definitely Setuju. This statement is a direct consequence of the function's amplitude. The value of '2' in f(x) = 2sin(x) influences this. Always remember the amplitude will be the difference between the max and min value divided by 2. You can also find the max value by looking at the coefficient in front of the sine. It must be 2.

Statement 3: The period of the function is π.

This statement asks us about the period of the function. Recall the period is the length of one complete cycle of the function, and for the basic sine function sin(x), this is 2π. When we change the amplitude, it vertically stretches or compresses the graph, but does not change its period. This means, the period is unchanged when you only modify the amplitude. Also, the coefficient in front of the x within the sine function will impact the period. Since there are no coefficients within the sine function other than the x, the period of f(x) = 2sin(x) is also 2π. Therefore, the statement is Tidak Setuju. Understanding the period is critical for graphing the sine function. Make sure you understand the impact different coefficients have on the period.

Statement 4: The function passes through the origin (0,0).

This one asks about the function's behavior at a specific point. The sine function sin(x), and therefore 2sin(x), starts at the origin. If you plug in x = 0 into f(x) = 2sin(x), you get f(0) = 2sin(0) = 20 = 0*. This means the graph does, in fact, pass through the point (0,0), where x = 0 and y = 0. Therefore, we Setuju with this statement. This statement highlights an important feature. The sine function's starting point is the origin. The amplitude and period affect the overall shape and scale, but they don't change where it starts. Knowing the properties of the sine function, we can easily identify this point. Always consider this point when sketching or thinking about the sine graph, it gives you a reference point and helps you visualize it more effectively.

Conclusion

So, there you have it, guys! We’ve carefully examined each statement related to f(x) = 2sin(x). Understanding the amplitude, period, and the basic behavior of the sine function is crucial for correctly answering these questions. Always visualize the graph in your head, or even sketch it out quickly, to help you analyze these statements! Keep practicing, and you'll become a trig master in no time! This analysis helps solidify your understanding of how the 2 in front of sin(x) affects the function's graph and its key properties.