Problem 24 Graph one full period of each fu... [FREE SOLUTION]

Chapter 4: Problem 24

Graph one full period of each function. $$y=\frac{3}{2} \cot \left(3 x+\frac{\pi}{4}\right)$$

Short Answer

Expert verified

The function $y=\frac{3}{2} \cot \left(3 x+\frac{\pi}{4}\right)$ has a period of $\frac{\pi}{3}$, phase shift of $-\frac{\pi}{12}$, and a vertical stretch of $ \frac{3}{2} $. The graph draws a 'U' that opens to left, beginning and ending with vertical asymptotes at $-\frac{\pi}{12}$ and $\frac{5\pi}{12}$, respectively, passing through $(\frac{\pi}{12}, 0)$.

Step by step solution

Identify the period and phase shift

Given the generalized form $ y = A \cot (Bx - C) $, where 'A' is the amplitude, 'B' is the period and 'C' is the phase shift, here $ A=\frac{3}{2} $, $ B=3 $, and $ C=-\frac{\pi}{4} $. The period of the cotangent function is $\frac{\pi}{B}$, so here it will be $\frac{\pi}{3}$. The phase shift is $\frac{C}{B}$, which gives $-\frac{\pi}{12}$. Remember the function shifts to the right if phase shift is negative.

Plot key points

Start by plotting the phase shift as the starting point, which is $-\frac{\pi}{12}$. For the cotangent function, important points within a period are at the phase shift (which is a vertical asymptote), half-period (which is zero) and full period (which is another asymptote). In this case, these are at $-\frac{\pi}{12}$ (asymptote), $\frac{\pi}{12}$ (zero), and $\frac{5\pi}{12}$ (asymptote). Plot these points.

Draw the general shape of cotangent function

Draw the general shape of the cotangent function starting from the asymptote ($-\frac{\pi}{12}$), going through the zero point ($\frac{\pi}{12}$), and ending at the other asymptote ($\frac{5\pi}{12}$). This shape should resemble a 'U' that opens to the left.

Apply vertical transformation

The function is vertically stretched by a factor of $\frac{3}{2}$. Every y-coordinate should be multiplied by this value. As only the zero-point lies on the y-axis, it won't move, so the stretch only influences the steepness of the curve.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Period of Cotangent Function

Understanding the period of the cotangent function is essential when graphing it. The period refers to the length of one complete cycle of the function before it starts repeating. For the cotangent function, which is represented as $ y = A \cot(Bx - C) $, the period is calculated using the formula $ \frac{\pi}{B} $.

In the case of $ y = \frac{3}{2} \cot(3x + \frac{\pi}{4}) $, we identify the value of 'B' as 3. Therefore, the period is $ \frac{\pi}{3} $, which means that after an interval of $ \frac{\pi}{3} $ radians, the cotangent function will repeat its pattern. Recognizing the period is the first step toward accurately graphing the function.

Phase Shift

The phase shift of a trigonometric function describes its horizontal displacement along the x-axis. For the function $ y = \frac{3}{2} \cot(3x + \frac{\pi}{4}) $, we determine the phase shift by the formula $ \frac{C}{B} $, using the constants from the function's standard form. In this case, 'C' is $ \frac{\pi}{4} $ and 'B' is 3, leading to a phase shift of $ -\frac{\pi}{12} $.

A negative phase shift indicates that the graph shifts to the right. This adjustment is crucial to place the function correctly on the coordinate system. By incorporating this shift, we ensure the function will start and end at the right points within a period.

Vertical Asymptote

The vertical asymptote of the cotangent function is a critical concept. An asymptote is a line the function approaches but never touches. For $ y = \frac{3}{2} \cot(3x + \frac{\pi}{4}) $, vertical asymptotes occur at values of 'x' where the cotangent function is undefined, namely when $ 3x + \frac{\pi}{4} $ is an integer multiple of $ \pi $.

The exercise reveals two vertical asymptotes within one period: one at $ -\frac{\pi}{12} $ and another at $ \frac{5\pi}{12} $. Recognizing these allows us to accurately illustrate the function since the curve approaches the asymptotes infinitely without crossing them.

Trigonometric Function Transformations

The last fundamental concept to grasp when graphing trigonometric functions is trigonometric function transformations. These transformations include scaling, shifting, and reflecting. For our function $ y = \frac{3}{2} \cot(3x + \frac{\pi}{4}) $, the coefficient $ \frac{3}{2} $ represents a vertical stretch. The graph of the function stretches away from the x-axis by a factor of $ \frac{3}{2} $ times the original value.

This transformation affects the steepness of the cotangent curve but does not alter its zeros or asymptotes. The understanding of how to apply these transformations ensures the graph reflects the altered amplitude accurately, providing a true representation of the function.

91影视

Graph one full period of each function. $$y=\frac{3}{2} \cot \left(3 x+\frac{\pi}{4}\right)$$

Short Answer

Step by step solution

Identify the period and phase shift

Plot key points

Draw the general shape of cotangent function

Apply vertical transformation

Key Concepts

Period of Cotangent Function

Phase Shift

Vertical Asymptote

Trigonometric Function Transformations

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