Analyzing The Periodic Function F(x) = 5cos(2x) - 1

Hey guys! Today, we're diving deep into analyzing the periodic function f(x) = 5cos(2x) - 1. We'll break down its key characteristics, including amplitude, period, and maximum values. So, let's jump right in and make sure we understand each aspect of this function. Understanding these concepts is super important for grasping more complex math and physics problems later on. So grab your thinking caps, and let's get started!

1. Understanding Amplitude

Let's kick things off by understanding the amplitude of our function, f(x) = 5cos(2x) - 1. The amplitude of a periodic function, especially a trigonometric one like cosine or sine, tells us about the height of the function's wave, or more precisely, the distance from its midline to its maximum or minimum point. It's essentially the measure of how much the function oscillates up and down from its central position.

In the given function, the general form we're looking at is f(x) = A cos(Bx) + C, where A represents the amplitude, B affects the period, and C is the vertical shift. For our specific function, f(x) = 5cos(2x) - 1, we can see that the coefficient in front of the cosine function is 5. This number, 5, is our A, and it directly corresponds to the amplitude of the function.

So, what does this mean in practical terms? It means that the cosine wave in our function is stretched vertically by a factor of 5. The basic cosine function, cos(x), oscillates between -1 and 1. However, when we multiply it by 5, the function 5cos(2x) now oscillates between -5 and 5. This vertical stretch gives the function its characteristic height. Therefore, it's correct to say that the amplitude of the function f(x) is indeed 5. When you look at the graph of the function, you'll visually see this oscillation spanning 5 units above and 5 units below the midline.

In simple terms, the amplitude is like the volume knob on a sound system – it controls how "loud" or "intense" the wave is. A larger amplitude means a more pronounced oscillation, while a smaller amplitude means a more subdued one. For f(x) = 5cos(2x) - 1, the amplitude of 5 tells us that the function has a significant vertical stretch, making its peaks and troughs quite prominent. Keep this in mind as we move on to exploring other characteristics of the function. Understanding the amplitude is crucial for visualizing and interpreting the behavior of periodic functions in various contexts.

2. Calculating the Period

Alright, let's tackle the next key characteristic of our periodic function: the period. The period, in simple terms, is the length of one complete cycle of the function. Think of it as the amount of time or distance it takes for the function to repeat its pattern. For trigonometric functions like cosine and sine, the period is a fundamental property that dictates how frequently the function oscillates. To figure out the period of f(x) = 5cos(2x) - 1, we need to focus on the part of the function that affects the horizontal stretch or compression, which is the coefficient of x inside the cosine function.

In the general form f(x) = A cos(Bx) + C, the period is determined by the value of B. Specifically, the period T is calculated using the formula T = 2π / |B|, where 2π is the natural period of the standard cosine function, cos(x), and |B| is the absolute value of B. This formula tells us how much the function's cycle is compressed or stretched compared to the standard cosine function. For our function, f(x) = 5cos(2x) - 1, the value of B is 2. This means that the argument inside the cosine function is 2x, which will affect how quickly the cosine wave completes one full cycle.

Now, let's plug B = 2 into our period formula: T = 2π / |2| = 2π / 2 = π. This calculation tells us that the period of the function f(x) = 5cos(2x) - 1 is π. So, the function completes one full cycle in an interval of length π. This is a crucial piece of information because it allows us to visualize how the function behaves over a specific range of x values. If we were to graph this function, we would see that the cosine wave repeats itself every π units along the x-axis.

Why is this important? Knowing the period helps us predict the behavior of the function beyond a single cycle. For instance, if we know the function's values in the interval [0, π], we know the function's values for all intervals [nπ, (n+1)π], where n is an integer. This makes analyzing and applying periodic functions much more manageable. Therefore, the statement the period of the function f(x) is π is correct. Understanding the period is like knowing the rhythm of a song – it tells you how often the melody repeats itself, making it easier to follow and predict.

3. Finding the Maximum Value

Okay, team, let's pinpoint where our function f(x) = 5cos(2x) - 1 hits its peak – that's right, we're talking about finding the maximum value! To nail this, we need to think about the cosine function's behavior. Remember, the basic cosine function, cos(x), swings between -1 and 1. This means its highest possible value is 1. Our function, f(x) = 5cos(2x) - 1, is a modified version of the cosine function, so we need to figure out how these modifications affect its maximum value.

First, let's tackle the 5cos(2x) part. As we discussed earlier, the amplitude of 5 stretches the cosine function vertically. So, 5cos(2x) will oscillate between -5 and 5. Therefore, the maximum value of 5cos(2x) is 5. Now, let's bring in the "- 1" part of our function. This is a vertical shift, which moves the entire function down by 1 unit. So, to find the maximum value of f(x) = 5cos(2x) - 1, we take the maximum value of 5cos(2x) (which is 5) and subtract 1. This gives us a maximum value of 5 - 1 = 4. Thus, the maximum value that our function f(x) can reach is 4. Knowing the maximum value is crucial because it gives us a sense of the upper bound of the function's range.

Now, the next part of our mission is to figure out where this maximum value occurs. The cosine function cos(2x) reaches its maximum value of 1 when its argument, 2x, is a multiple of 2π, meaning 2x = 2πn, where n is an integer. If we solve for x, we get x = πn. So, the cosine part of our function hits its peak at x = 0, π, 2π, and so on. This means that 5cos(2x) will be at its maximum value of 5 at these points. Since our full function is f(x) = 5cos(2x) - 1, it will reach its maximum value of 4 at the same x values where 5cos(2x) is at its maximum.

Therefore, the statement the function reaches its maximum value at x = 0 is correct because when x = 0, f(0) = 5cos(0) - 1 = 5(1) - 1 = 4, which is the maximum value. In conclusion, by understanding how the amplitude, vertical shift, and the nature of the cosine function interact, we've successfully identified both the maximum value of the function and where it occurs. This kind of analysis is super handy for tackling more complex problems in trigonometry and calculus.

Conclusion

Wrapping things up, guys, we've dissected the function f(x) = 5cos(2x) - 1 like pros! We confirmed that its amplitude is 5, meaning it oscillates 5 units above and below its midline. We nailed down the period as π, which tells us the function completes one full cycle within this interval. And, we discovered that the function hits its maximum value of 4 at x = 0. These insights give us a solid understanding of how this periodic function behaves.

Understanding these concepts is crucial for anyone diving into mathematics, physics, or engineering. Knowing how to analyze functions like this helps us predict and model real-world phenomena, from waves in the ocean to the oscillations in electrical circuits. So, keep practicing and exploring different functions – you're building a fantastic foundation for more advanced studies! Keep up the great work, and remember, math can be fun when you break it down step by step. You got this!