Problem 43 - Precalculus : Building Concepts and Connections

Question

Graph the given pair of functions in the same window. Graph at least two cycles of each function, and describe the similarities and differences between the graphs. $$f(x)=\cot (3 \pi x) ; f(x)=\cot \left(\frac{\pi}{3} x\right)$$

Step-by-Step Solution

Answer

Both cotangent functions oscillate from positive to negative infinity. f(x) = cot(3πx) oscillates every 1/3 units while f(x) = cot(π/3x) oscillates every 3 units. The main difference between these functions is their frequency of oscillation within a given interval, determined by their respective period.

Step 1: Recognize and calculate the period of each function

Ordinarily, f(x) = cot(x) has a period of \(\pi\), that is, the function repeats after an interval of \(\pi\). However, when multiplied by a constant as seen in both functions, this changes the period. For f(x) = cot(3πx), this creates a period of \(\pi/(3\pi)\) = \(1/3\). On the other hand, f(x) = cot(π/3x) yields a period of \(\pi/(π/3)\) = 3.

Step 2: Draw the graphs for each function

To graph these functions create a graph where the x-axis represents the values of x and the y-axis represents the function's value comprising both positive and negative infinities. The graph of f(x)= cot(3πx) will present an oscillation between positive infinity and negative infinity every 1/3 units. As for f(x) = cot(π/3x), the oscillation will occur every 3 units. Indicating these oscillations on the graph will result in a series of vertical asymptotes with the cotangent function oscillating between.

Step 3: Analyze the similarities and differences

Similarities between the two graphs lie in their oscillation pattern and range from positive to negative infinity. The differences, on the other hand, exist in their periodicity. f(x) = cot(3πx) has a shorter period (1/3) and therefore oscillates more frequently within a given interval compared to f(x) = cot(π/3x).

Key Concepts

Periodicity in Trigonometric Functions

Periodicity is a key feature in trigonometric functions. Typically, it represents how long it takes for a function to repeat its values. For the standard cotangent function, the period is \(\pi\). However, when the argument of the function is multiplied by a constant, the period changes.

In the case of \(f(x) = \cot(3\pi x)\), the factor \(3\pi\) affects how quickly the function repeats. The new period becomes \(\frac{\pi}{3\pi} = \frac{1}{3}\). Essentially, this function completes a full cycle in just \(1/3\) of the time it normally would.

Alternatively, \(f(x) = \cot\left(\frac{\pi}{3} x\right)\) has its period calculated by \(\frac{\pi}{\frac{\pi}{3}} = 3\). Therefore, this function takes a longer time to complete one cycle—3 units. Understanding periodicity helps make sense of a trigonometric function's long-term behavior.

Understanding the Cotangent Function

The cotangent function, denoted as \(\cot(x)\), is the reciprocal of the tangent function: \(\cot(x) = \frac{1}{\tan(x)}\). This function has some unique characteristics. It is undefined at points where tangent is zero, which occurs at multiples of \(\pi\), leading to vertical asymptotes in the graph.

The cotangent function oscillates between positive and negative infinity around these asymptotes. Because \(\cot(x)\) never touches an x-value where tan(x) equals zero, we see these distinct breaks in the cotangent graph.

Both graphs, \(f(x) = \cot(3\pi x)\) and \(f(x) = \cot\left(\frac{\pi}{3} x\right)\), present this typical cotangent behavior, but at different periodic intervals. The faster oscillation in \(f(x) = \cot(3\pi x)\) compared to \(f(x) = \cot\left(\frac{\pi}{3} x\right)\) illustrates how altering the period changes the function's frequency of oscillation.

Graphing Trigonometric Functions

Graphing trigonometric functions like the cotangent can be visually insightful. On a standard coordinate plane, the x-axis typically represents the angle \(x\), while the y-axis represents the cotangent value. When graphing \(f(x) = \cot(3\pi x)\) and \(f(x) = \cot\left(\frac{\pi}{3} x\right)\), you are essentially plotting how these functions behave over their respective periods.

- **Vertical Asymptotes**: These appear where the cotangent function is undefined. For \(f(x) = \cot(3\pi x)\), these asymptotes occur every 1/3 unit causing frequent breaks. For \(f(x) = \cot\left(\frac{\pi}{3} x\right)\), these appear every 3 units.- **Oscillation Patterns**: These functions fluctuate between positive and negative infinity in their defined range. This movement is what creates the familiar wave-like pattern.- **Similarities and Differences**: While both share the same range and oscillation behavior, the period difference means \(f(x) = \cot(3\pi x)\) oscillates more frequently within any interval compared to \(f(x) = \cot\left(\frac{\pi}{3} x\right)\).
By understanding these patterns, students can better interpret the periodic nature and transformation of trigonometric functions on a graph.