Problem 21 Graph two periods of the given c... [FREE SOLUTION]

Chapter 4: Problem 21

Graph two periods of the given cotangent function. $$y=-3 \cot \frac{\pi}{2} x$$

Short Answer

Expert verified

The graph of the function $y=-3 \cot \left(\frac{\pi}{2} x\right)$ will have a period of 4 and an amplitude of 3, starting from negative infinity at the multiples of the period (x=0,4, and 8), passing through the x-axis at the quarter and three-quarters of the period (x=1, 3, 5, and 7), and reaching positive infinity at the half of the period (x=2 and 6).

Step by step solution

Identify the Amplitude and Period

In the given function, the amplitude (height) of the function is given by the absolute value of the multiplier in front of the cotangent function, which is 3 in this case. Changes in amplitude do not affect the period or frequency of the function, but they do affect the maximum and minimum values of the function. The period (interval over which the function repeats) is given by $\frac{2\pi} {\text{coefficient of x inside the cotangent function}} = 4$. Plasma has a period of 4.

Plot the Key Points

We know that a cotangent function starts at positive infinity, drops to 0, and then goes to negative infinity within its period. So, at x=0, the function will start at negative infinity (since our function is -cot(x)). After a quarter period (1), the function will reach its trough (0). After half the period (2), it will climb back up to positive infinity. After three-quarters of the period (3), it will pass through 0 again, and at the end of the period (4), it will return to negative infinity. These points will provide the key shape of the graph. Since two periods are required, repeat the pattern for the next period from x=4 to x=8.

Draw the Asymptotes

Asymptotes are the boundaries of the function and occur at the points where the function is undefined. So, we will draw vertical asymptotic lines at every multiple of the period (x=0, 4, 8).

Draw the Graph

The graph will start from negative infinity at x=0, cross through the x-axis at x=1, go to positive infinity at x=2, cross the x-axis again at x=3, and return to negative infinity at x=4. This pattern repeats for the next period.

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Key Concepts

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

Amplitude

In the cotangent function, the term amplitude refers to the vertical stretch or compression of the graph. It is identified by the absolute value of the multiplier in front of the trigonometric function, which in this case is 3. This means that the graph will be taller or more pronounced compared to a standard cotangent function. While amplitude changes the height of the graph, it does not influence other characteristics like the period or frequency of the function. Amplitude only affects how steep or shallow the graph appears.

Period

The period of a cotangent function determines the interval over which the function repeats its pattern. For the function $y=-3 \cot \frac{\pi}{2} x$, the period can be calculated by dividing $2\pi$ by the coefficient of $x$ in the cotangent function. This gives us $\frac{2\pi}{\frac{\pi}{2}} = 4$. Therefore, the function has a period of 4, meaning the complete wave pattern repeats every 4 units along the x-axis. This understanding helps in predicting the behavior of the graph and where it will end and begin its repetition.

Asymptotes

Asymptotes are essential elements in the graph of a cotangent function. They represent vertical lines where the function is undefined and which the graph approaches but never touches. For the given function, asymptotes occur at intervals matching the period, specifically at every multiple of 4 (e.g., $x = 0, 4, 8$). It is important to draw these lines when graphing the cotangent function, as they help define the areas where the function dives into infinity, thus acting as imaginary boundaries guiding the curve's path.

Trigonometric Functions

Trigonometric functions, including the cotangent, are mathematical functions that relate angles to ratios of a right triangle's sides. These functions are periodic, meaning they repeat their values in regular intervals. The cotangent of an angle, $\cot(x)$, is the reciprocal of the tangent; it equals $\frac{1}{\tan(x)}$, or $\frac{\cos(x)}{\sin(x)}$. In graph form, it shows oscillations that start at infinity, decrease to zero, and go back to negative infinity over a period. Understanding these functions helps in analyzing their graphs and predicting their behaviors in various contexts.

91影视

Graph two periods of the given cotangent function. $$y=-3 \cot \frac{\pi}{2} x$$

Short Answer

Step by step solution

Identify the Amplitude and Period

Plot the Key Points

Draw the Asymptotes

Draw the Graph

Key Concepts

Amplitude

Period

Asymptotes

Trigonometric Functions

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