Let's dive deep into the intricacies of the function f(x) = -4 cot(4x). To find the zeros of this cotangent function, we must first grasp the fundamental properties of the cotangent itself. The cotangent function, denoted as cot(x), is defined as the ratio of the cosine to the sine function: cot(x) = cos(x) / sin(x). A crucial aspect of the cotangent function is its periodicity and the locations where it becomes zero. Cotangent function zeros occur where the cosine function is zero, which is at odd multiples of π/2 (i.e., π/2, 3π/2, 5π/2, and so on). Furthermore, the cotangent function has vertical asymptotes where the sine function equals zero, corresponding to integer multiples of π (i.e., 0, π, 2π, etc.). Understanding these characteristics is vital for identifying the zeros of any cotangent-related function.
Now, let's consider the transformation applied to the cotangent function in our given equation, f(x) = -4 cot(4x). The coefficient -4 in front of the cotangent term represents a vertical stretch and a reflection across the x-axis, but it does not affect the zeros of the function. The argument 4x inside the cotangent function, however, significantly impacts the zeros. This transformation compresses the graph horizontally by a factor of 4. This means the zeros, which originally occurred at odd multiples of π/2, will now occur at values of x such that 4x equals those odd multiples of π/2. Thus, the zeros of cot(4x) will be at x = (π/2 + nπ)/4, where n is an integer. Simplifying this expression gives us x = π/8 + nπ/4. This foundational knowledge sets the stage for a systematic approach to determine which of the provided expressions correctly identifies all the zeros of the function f(x) = -4 cot(4x).
In the subsequent sections, we will methodically analyze each of the given expressions, comparing them against our derived understanding of the function's zeros. This involves substituting integer values for n into each expression and examining the resulting values of x. By contrasting these results with the expected pattern of zeros for cot(4x), we can pinpoint the expression that accurately represents all the zeros of the function. Moreover, this exercise highlights the significance of understanding function transformations and their influence on the roots of trigonometric functions. The horizontal compression caused by the 4x argument plays a pivotal role, and any misinterpretation of this effect can lead to the incorrect identification of the zeros. Therefore, a careful, step-by-step analysis is essential to ensure that the correct expression is chosen, solidifying our understanding of how trigonometric functions behave under various transformations. This comprehensive exploration will not only provide the correct answer but also enhance our problem-solving skills in dealing with trigonometric functions and their zeros.
In this section, we will methodically evaluate each of the provided expressions to ascertain which one accurately represents the zeros of the function f(x) = -4 cot(4x). Our primary goal is to find an expression that, when integer values are substituted for n, yields all possible zeros of the given function. This involves a detailed analysis of each option, comparing the resulting values with the expected pattern of zeros for a cotangent function that has undergone horizontal compression.
Option a: π/8 + 4nπ
Let's start by examining the first expression, π/8 + 4nπ. This expression suggests that the zeros occur at π/8 plus integer multiples of 4π. To test its validity, we can substitute different integer values for n and observe the resulting values of x. When n = 0, we get x = π/8. When n = 1, we get x = π/8 + 4π = 33π/8. When n = -1, we get x = π/8 - 4π = -31π/8. These values indicate that the zeros are spaced far apart, with intervals of 4π between them. However, this spacing does not align with our understanding of the function f(x) = -4 cot(4x). We know that the 4x term inside the cotangent function compresses the zeros, leading to a more frequent occurrence of zeros than what this expression suggests. Therefore, based on this analysis, expression a does not appear to correctly represent all the zeros of the function.
Option b: π/2 + 4nπ
Next, we consider the expression π/2 + 4nπ. This option implies that the zeros occur at π/2 plus integer multiples of 4π. Similar to our previous approach, we substitute various integer values for n to observe the pattern of zeros. When n = 0, we get x = π/2. When n = 1, we get x = π/2 + 4π = 9π/2. When n = -1, we get x = π/2 - 4π = -7π/2. This expression, like the first one, generates zeros that are spaced at intervals of 4π, which is a wide gap. The function f(x) = -4 cot(4x), with its horizontal compression, should have zeros occurring much more frequently. Thus, this expression also seems unlikely to be the correct representation of the function's zeros. The large intervals between the zeros suggest that this expression misses the crucial effect of the horizontal compression caused by the 4x term in the cotangent function. Therefore, after careful examination, it is evident that this option does not accurately capture the distribution of zeros for the given function.
Option c: 2π + n(π/4)
Now, let's delve into the expression 2π + n(π/4). This option suggests that the zeros occur at 2π plus integer multiples of π/4. We follow our established methodology of substituting different integer values for n to observe the resulting pattern of zeros. When n = 0, we get x = 2π. When n = 1, we get x = 2π + π/4 = 9π/4. When n = 2, we get x = 2π + 2(π/4) = 5π/2. When n = -1, we get x = 2π - π/4 = 7π/4. These values indicate that the zeros are spaced at intervals of π/4, but they are all offset by an initial value of 2π. While the interval spacing of π/4 aligns with the compression factor in cot(4x), the initial offset of 2π is concerning. The fundamental period of cot(4x) is π/4, and its zeros should not be offset by such a significant amount. This observation raises doubts about the correctness of this expression. The consistent spacing of π/4 suggests that the expression captures the compression effect of the 4x term, but the 2π offset indicates a potential misinterpretation of the function's fundamental behavior. Therefore, this option warrants further scrutiny and comparison with the function's expected behavior.
Option d: π/8 + n(π/4)
Finally, we consider the expression π/8 + n(π/4). This option proposes that the zeros occur at π/8 plus integer multiples of π/4. Following our established procedure, we substitute integer values for n to observe the resulting pattern. When n = 0, we get x = π/8. When n = 1, we get x = π/8 + π/4 = 3π/8. When n = 2, we get x = π/8 + 2(π/4) = 5π/8. When n = -1, we get x = π/8 - π/4 = -π/8. This expression generates a series of zeros spaced at intervals of π/4, starting from π/8. This pattern precisely matches our derived understanding of the zeros for cot(4x). We recall that the zeros of cot(x) occur at odd multiples of π/2, and the 4x term compresses the function horizontally by a factor of 4. Therefore, the zeros of cot(4x) should occur at x = (π/2 + nπ)/4 = π/8 + nπ/4, which is exactly what this expression represents. The consistent spacing of π/4 and the starting point of π/8 perfectly align with the expected behavior of the function. Consequently, this expression appears to be the correct representation of the zeros of f(x) = -4 cot(4x).
After meticulously evaluating each of the provided expressions, we can now definitively determine the one that accurately represents the zeros of the function f(x) = -4 cot(4x). Our analysis involved substituting integer values for n in each expression and comparing the resulting pattern with our understanding of the cotangent function's behavior, particularly its horizontal compression due to the 4x term.
Expressions a (π/8 + 4nπ) and b (π/2 + 4nπ) were quickly ruled out because they produced zeros that were spaced too far apart. The interval of 4π between the zeros contradicted the horizontal compression caused by the 4x term, which should lead to a more frequent occurrence of zeros. These expressions failed to capture the fundamental characteristic of the transformed cotangent function, making them unsuitable representations of the function's zeros. The wide spacing indicated a significant deviation from the expected behavior, highlighting the importance of considering the transformation's impact on the zeros.
Expression c (2π + n(π/4)) showed a spacing of π/4 between the zeros, which aligned with the compression factor. However, the initial offset of 2π was a point of concern. The zeros of cot(4x) should not have such a significant offset, given the function's period. While this expression captured the correct interval, the offset suggested a potential misunderstanding of the function's fundamental behavior. The offset raised questions about the expression's overall accuracy, prompting a closer examination of the function's properties.
Expression d (π/8 + n(π/4)), on the other hand, perfectly matched our derived understanding of the function's zeros. It produced a series of zeros spaced at intervals of π/4, starting from π/8. This pattern aligned precisely with the zeros of cot(4x), which we determined to be at x = (π/2 + nπ)/4 = π/8 + nπ/4. The expression correctly captured both the spacing and the starting point of the zeros, making it the most accurate representation. The perfect alignment with the expected behavior solidified its position as the correct expression.
Therefore, based on our comprehensive evaluation, the expression that gives all of the zeros of the function f(x) = -4 cot(4x) is d) π/8 + n(π/4). This expression accurately accounts for the horizontal compression caused by the 4x term and correctly represents the periodic nature of the cotangent function.
In conclusion, to determine the zeros of the function f(x) = -4 cot(4x), we undertook a systematic and detailed analysis. We began by understanding the fundamental properties of the cotangent function, including its periodicity and the location of its zeros and asymptotes. We then considered the transformations applied to the cotangent function, particularly the horizontal compression caused by the 4x term. This understanding allowed us to derive the expected pattern of zeros for the given function.
Next, we meticulously evaluated each of the provided expressions by substituting integer values for n and comparing the resulting patterns with our expectations. Expressions a and b were quickly eliminated due to the wide spacing between their zeros, which contradicted the horizontal compression. Expression c, while having the correct spacing, had an offset that raised concerns. Only expression d, π/8 + n(π/4), perfectly matched the expected pattern of zeros for the function f(x) = -4 cot(4x).
Our analysis highlights the importance of a thorough understanding of trigonometric functions and their transformations. The horizontal compression caused by the 4x term played a crucial role in determining the zeros, and any misinterpretation of this effect could have led to an incorrect answer. By carefully evaluating each expression and comparing it with our derived understanding, we confidently identified the correct representation of the function's zeros.
This exercise demonstrates a robust approach to solving mathematical problems involving trigonometric functions. It emphasizes the need for a step-by-step analysis, careful consideration of function transformations, and a comparison of results with expected behavior. The process not only provides the correct answer but also enhances problem-solving skills and deepens our understanding of mathematical concepts. The accurate determination of the zeros for f(x) = -4 cot(4x) underscores the effectiveness of a systematic approach and a solid grasp of trigonometric principles.