Problem 78 - Precalculus

Question

For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes. $$ f(x)=-3 \cot (2 x) $$

Step-by-Step Solution

Answer

The stretching factor is -3, the period is \( \frac{\pi}{2} \), and asymptotes are at \( x = \frac{k\pi}{2} \), where \( k \) is an integer.

Step 1: Identify the Basic Function

The function given is \( f(x) = -3 \cot(2x) \). Start by recognizing the basic function, which is \( \cot(x) \), the cotangent function. This function has a period of \( \pi \) and vertical asymptotes at multiples of \( \pi \).

Step 2: Determine the Stretching Factor

The function \( f(x) = -3 \cot(2x) \) has a coefficient in front of the cotangent function. The coefficient \(-3\) is the stretching factor, which vertically stretches or compresses the graph and also reflects it over the x-axis.

Step 3: Calculate the New Period

The given function involves \( \cot(2x) \). To find the period of \( \cot(2x) \), use the formula for cotangent new period = \( \frac{\pi}{|b|} \), where \( b = 2 \). Therefore, the new period is \( \frac{\pi}{2} \).

Step 4: Determine the Asymptotes

The original \( \cot(x) \) has asymptotes at \( x = k\pi \), where \( k \) is an integer. For \( \cot(2x) \), asymptotes occur every \( \frac{\pi}{2} \). Thus, the vertical asymptotes are at \( x = \frac{k\pi}{2} \) for integers \( k \).

Step 5: Sketch the Graph

To sketch two periods of \( f(x) = -3 \cot(2x) \), plot the graph within the domain of \( -\pi \) to \( \pi \). Mark asymptotes at \( x = -\frac{3\pi}{2}, -\pi, -\frac{\pi}{2}, 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \). The graph will have downward opening branches between each pair of asymptotes.

Key Concepts

Cotangent Function

The cotangent function, represented as \( \cot(x) \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the tangent function:

• \( \cot(x) = \frac{1}{\tan(x)} \)
• Or equivalently, \( \cot(x) = \frac{\cos(x)}{\sin(x)} \)

This means that whenever the tangent of \( x \) equals zero, the cotangent is undefined. The graph of the cotangent function is characterized by its

• periodic nature,
• oscillating between positive and negative infinity,
• with a repeating cycle.

For the basic cotangent function, \( \cot(x) \), the period is \( \pi \), meaning the pattern of the graph repeats every \( \pi \) units. Vertical asymptotes appear wherever the sine of \( x \) is zero, which occurs at multiples of \( \pi \), resulting in undefined points that lead to these asymptotes.
In the function \( f(x) = -3 \cot(2x) \), this idea is kept but transformed through stretching and period modification.

Period of Functions

The period of a trigonometric function is the length over which the function's shape repeats. Calculating the period of transformed trigonometric functions can be accomplished using the form:

• \( \text{Period} = \frac{\pi}{|b|} \) for the cotangent function \( \cot(bx) \)

For \( f(x) = -3 \cot(2x) \), the factor \( b = 2 \) changes the period from the original \( \pi \) to \( \frac{\pi}{2} \). This means the graph will complete its cycle quicker, allowing more cycles within any given domain. The negative sign doesn't affect the period but reflects the graph downwards around the x-axis.
Understanding how each part of a function impacts its period helps predict the graph's shape. It's a vital tool when sketching transformations of trigonometric functions, especially under different coefficients.

Vertical Asymptotes

Vertical asymptotes are straight vertical lines that the graph approaches but never touches or intersects. They occur in functions where division by zero happens. For the cotangent function \( \cot(x) \) and any of its transformations like \( \cot(bx) \), these asymptotes occur where sine equals zero, since \( \cot(x) = \frac{\cos(x)}{\sin(x)} \).
For \( \cot(x) \), vertical asymptotes are at each integer multiple of \( \pi \), such as \( x = 0, \pi, 2\pi, \ldots \). However, when the cotangent function is transformed, as in \( \cot(2x) \), the position of these asymptotes is also transformed based on the period adjustment.

• For \( f(x) = -3 \cot(2x) \), the vertical asymptotes are found at \( x = \frac{k\pi}{2} \), where \( k \) is an integer.

Finding vertical asymptotes is crucial when plotting the graph manually because it tells you where the function will approach infinity and gives a clearer picture of its overall behavior.