Problem 18 - Algebra and Trigonometry

Question

In Exercises 17–24, graph two periods of the given cotangent function. $$ y=\frac{1}{2} \cot x $$

Step-by-Step Solution

Answer

The graph of \(y = \frac{1}{2} \cot x\) is a cotangent curve, vertically stretched by a factor of \(\frac{1}{2}\) and repeating every \(\pi\) units along the x-axis.

Step 1: Identification

The function given is \(y = \frac{1}{2} \cot x\). This is a transformed cotangent function, with vertical stretch by a factor of \(\frac{1}{2}\). The base cotangent function has period of \(\pi\).

Step 2: Determine key characteristics

The cotangent function has asymptotes at \(0, \pi, 2\pi\), etc. Since the function is not translated or dilated horizontally, the asymptotes for the given function are also at \(0, \pi, 2\pi\), etc. The function is positive in the interval \((0, \pi)\) and negative in the interval \((\pi, 2\pi)\).

Step 3: Plot the function

Start by drawing asymptotes at \(x = 0\), \(x = \pi\) and \(x = 2\pi\). Between each pair of asymptotes, plot a cotangent curve, dipping at the midpoint. Remember, the entire curve is vertically stretched by a factor of \(\frac{1}{2}\), meaning it dips to \(\frac{1}{2}\) and rises to \(-\frac{1}{2}\) at each period.

Key Concepts

Graphing Trigonometric Functions

When graphing trigonometric functions like the cotangent, it's essential to understand their basic shapes and behaviors. The cotangent function, \( \cot(x) \), is a cousin of the tangent function and shares some similarities in its graph. However, it has its unique features. For the cotangent function:

• Each period of cotangent extends from one vertical asymptote to the next, essentially filling the space between these lines.
• Unlike sine and cosine functions, cotangent doesn't have a maximum or minimum point within one period.
• It intersects the y-axis at points of zero function value, meaning it passes through zeros evenly distributed along the x-axis.

In our specific exercise, we graph two full periods of \( y = \frac{1}{2} \cot(x) \), recognizing that each period starts and ends at these asymptotes \( x = 0, \pi, 2\pi \), and so forth. By graphing segments between \( 0 \) and \( 2\pi \), we can visualize the entire function consistently.

Periodicity of Trigonometric Functions

Trigonometric functions are renowned for their periodic behavior, meaning they repeat their values over regular intervals. The cotangent function, \( \cot(x) \), has a period of \( \pi \), which means every \( \pi \) units along the x-axis, the function values repeat. This shorter period compared to the sine function's \( 2\pi \) is due to the cotangent's defining characteristics of asymptotes and zeros.
Understanding periodicity is crucial in plotting trigonometric functions as it tells us where key features like asymptotes and zeros will repeat. In our example, each full cycle of the cotangent function starts and ends at its vertical asymptotes.

• If you divide the cotangent function's period into four equal parts, it helps pinpoint essential points for sketching the curve.
• This understanding aids predicting behavior beyond the initial cycles we graph.

Seeing these regular repetitions allows for more straightforward graphing and aids the comprehension of how transformations impact the function's behavior.

Vertical Stretch Transformation

Vertical stretching occurs when the amplitude of a trigonometric function is modified, affecting how 'tall' or 'short' the function appears on the graph. For the transformed function \( y = \frac{1}{2} \cot(x) \), the vertical stretch factor is \( \frac{1}{2} \).
This means each value of the standard cotangent graph is halved, affecting the peaks, troughs, and scale of the graph. Here are some significant effects of this transformation:

• The graph appears to "squash" vertically, reducing the height of the peaks and troughs.
• All vertical distances from the x-axis to any point on the graph, whether above or below, are reduced by half. This provides a more compressed view of the typical cotangent wave.
• Even with this stretching, the asymptotes remain unchanged, as their location depends on the function's horizontal cycle, not its amplitude.

Understanding how a vertical stretch affects your trigonometric graph ensures a clearer, more accurate sketch that aligns with the transformation's impact on the function.