Analyzing F(x) = 3cos(2x) + 1 Amplitude, Period, Graph, And Solutions

In this detailed guide, we'll dive deep into the trigonometric function f(x) = 3cos(2x) + 1, exploring its properties, graphical representation, and solutions. This function combines the cosine function with transformations, making it a fascinating subject for mathematical analysis. Whether you're a student grappling with trigonometry or simply a math enthusiast, this article will provide a clear and comprehensive understanding of f(x).

Understanding the Function f(x) = 3cos(2x) + 1

To truly understand the function, let's break it down piece by piece. The function f(x) = 3cos(2x) + 1 is a transformation of the basic cosine function, cos(x). Several key parameters influence its behavior: the amplitude, the period, and the vertical shift. In this section, we'll dissect these parameters and understand their impact on the graph and characteristics of the function. The amplitude determines the maximum displacement from the midline, the period dictates how often the function repeats itself, and the vertical shift moves the entire graph up or down. Understanding these elements is crucial for sketching the graph and solving related problems.

Determining Amplitude and Period

Let's start by identifying the amplitude of the trigonometric function. The amplitude is the vertical distance from the midline (the horizontal line that runs through the middle of the graph) to the maximum or minimum point of the function. In our case, the function is f(x) = 3cos(2x) + 1. The coefficient of the cosine term, which is 3, determines the amplitude. Therefore, the amplitude of f(x) is 3. This means the graph will oscillate 3 units above and 3 units below its midline. The amplitude plays a significant role in determining the vertical stretch of the graph, making it a crucial parameter to understand.

Next, we need to figure out the period of the trigonometric function. The period is the horizontal distance it takes for the function to complete one full cycle. For a standard cosine function, cos(x), the period is 360 degrees (or 2π radians). However, the 2x inside the cosine function affects the period. To find the period of f(x) = 3cos(2x) + 1, we divide the standard period (360 degrees) by the coefficient of x, which is 2. So, the period of f(x) is 360° / 2 = 180 degrees. This means the function will complete one full cycle in 180 degrees, making it oscillate twice as fast as the standard cosine function. Understanding the period is essential for accurately plotting the graph and analyzing the function's cyclical behavior.

Visualizing the Graph of y = f(x)

Now that we know the amplitude and period, we can proceed to sketching the graph of y = f(x) = 3cos(2x) + 1. Graphing trigonometric functions might seem daunting at first, but with a systematic approach, it becomes quite manageable. We'll leverage our understanding of the amplitude, period, and vertical shift to create an accurate representation of the function's behavior. By identifying key points and connecting them smoothly, we can visualize the oscillating nature of the cosine function and its transformations.

First, let's consider the key points of the cosine function. A standard cosine function, cos(x), starts at its maximum value, goes down to its minimum value, and then returns to its maximum value over one period. In our case, the period is 180 degrees, so we'll focus on the interval from 0° to 180°. We can divide this interval into four equal parts: 0°, 45°, 90°, 135°, and 180°. These points will help us plot the critical values of the function.

Next, we need to account for the transformations. The amplitude of 3 means the function will oscillate between 1 + 3 = 4 and 1 - 3 = -2. The vertical shift of +1 moves the entire graph up by 1 unit. So, the midline of our function is y = 1. We can now plot the key points: at 0°, the function starts at its maximum value, which is 4; at 45°, it crosses the midline at 1; at 90°, it reaches its minimum value of -2; at 135°, it crosses the midline again at 1; and at 180°, it returns to its maximum value of 4. By connecting these points with a smooth curve, we create one full cycle of the graph. Since the function is defined for 0° ≤ x ≤ 360°, we simply repeat this cycle to cover the entire interval. This repetition showcases the periodic nature of the function, reinforcing the importance of understanding the period.

Determining the Number of Solutions

Finally, let's tackle the question of determining the number of solutions to the equation f(x) = k for some constant k. This involves analyzing the graph we've sketched and identifying the points where the function's value equals k. The number of solutions corresponds to the number of times the horizontal line y = k intersects the graph of y = f(x). This provides a visual and intuitive way to understand the solutions of trigonometric equations.

The number of solutions depends on the value of k. If k is outside the range of the function (i.e., k > 4 or k < -2), there are no solutions because the horizontal line y = k will not intersect the graph. If k is equal to the maximum or minimum value (i.e., k = 4 or k = -2), there will be two solutions within the interval 0° ≤ x ≤ 360° because the horizontal line y = k will touch the graph at its peaks or troughs. For any value of k between the maximum and minimum values (i.e., -2 < k < 4), there will be four solutions because the horizontal line y = k will intersect the graph at four distinct points. This analysis highlights the relationship between the graphical representation of the function and the algebraic solutions of equations involving the function.

Conclusion

In conclusion, the function f(x) = 3cos(2x) + 1 provides a rich example of trigonometric transformations. By understanding the amplitude, period, and vertical shift, we can accurately sketch its graph and analyze its behavior. The number of solutions to the equation f(x) = k can be determined by examining the intersections of the graph with the horizontal line y = k. This exploration demonstrates the power of graphical analysis in solving trigonometric problems and provides a solid foundation for further studies in mathematics. Keep practicing, guys, and you'll become trigonometric masters in no time!