Problem 24 - Precalculus : Building Concepts and Connections

Question

Use your knowledge of horizontal stretches and compressions to graph at least two cycles of the given functions. $$f(x)=\cos \left(\frac{1}{2} x\right)$$

Step-by-Step Solution

Answer

The function \(g(x) = \cot(\frac{\pi}{2} x)\) has a period of \(4\) caused by the horizontal compression factor of \(\frac{\pi}{2}\). It is undefined at \(x = 0\), \(x = 4\), \(x = 8\) etc., while it equals \(0\) at \(x = 2\), \(x = 6\), \(x = 10\) etc. and alternates between positive and negative infinity for each period.

Step 1: Understand the Cotangent Function

The cotangent function in its basic form, i.e., \(\cot(x)\), is periodic and repeats every interval of \(\pi\). This function approaches infinity at \(x = n\pi\; [n \in \mathbb{Z}]\) and is zero at \(\frac{\pi}{2} + n\pi\; [n \in \mathbb{Z}]\).

Step 2: Apply Horizontal Compression Factor

The function given is \(g(x) = \cot(\frac{\pi}{2} x)\). Here, the argument of the cotangent function is \(\frac{\pi}{2} x\). This introduces a horizontal compression by a factor of \(\frac{\pi}{2}\), so the function now repeats every \(\frac{2\pi}{\frac{\pi}{2}}\), or every \(4\) units along the x-axis.

Step 3: Plotting the Graph

Use the above information to construct the graph. Start by identifying a few critical points. For instance, one knows the function is undefined at \(x = 0\), \(x = 4\), \(x = 8\), etc. and equal to \(0\) at \(x = 2\), \(x = 6\), \(x = 10\) etc. The graph will alternate between positive and negative infinity at these points creating a series of spikes giving us two cycles.

Key Concepts

Cotangent Function

The cotangent function, denoted by \(\cot(x)\), is the reciprocal of the tangent function. This means \(\cot(x) = \frac{1}{\tan(x)}\). Its basic shape features a set of distinctive spikes. These occur where the function is undefined. Through every full period, \(\cot(x)\) starts at positive infinity, crosses zero, and approaches negative infinity.

• For \(\cot(x)\), the function's period is \(\pi\), meaning it repeats its pattern every \(\pi\) units along the x-axis.
• At integer multiples of \(\pi\), \(x = n\pi\), the cotangent function \((n \in \mathbb{Z})\) approaches positive or negative infinity.
• For values of \(x\) that are half-way through the period, \(\cot(x)\) crosses zero: \(x = \frac{\pi}{2} + n\pi\).

Understanding these key points allows one to plot the graph effectively by identifying where the zeroes and asymptotes lie.

Periodic Functions

Periodic functions repeat their values in regular intervals or periods. This feature makes them predictable over time. Trigonometric functions like sine, cosine, and cotangent are classic examples of periodic functions. For the cotangent function:

• The standard period is \(\pi\), indicating the amount of horizontal space needed for the pattern to repeat fully.
• The pattern includes rapid changes from positive to negative.
• This repetition allows for easier graphing when transformations, like horizontal compressions, are applied.

The concept of periodicity helps when modifying a function, since each transformation makes the intervals and repeating patterns manageable and consistent.

Horizontal Transformations

Horizontal transformations involve stretching or compressing a graph along the x-axis. They change the period of trigonometric functions. The general form of a trig function after a horizontal transformation can be expressed as \( f(kx) \), where \( k \) is the factor of transformation.

• If \( k > 1 \), there is a horizontal compression. The period decreases, making the function graph more condensed.
• If \( 0 < k < 1 \), there is a horizontal stretch, extending the period and spreading out the graph farther along the x-axis.

In the exercise \( g(x) = \cot(\frac{\pi}{2} x) \), the factor is \( \frac{\pi}{2} \). This compresses the graph of the cotangent function, changing its period from \( \pi \) to 4. As a result, the graph completes its cycle more quickly, every 4 units instead of every \( \pi \) units.