Problem 51
Question
In Exercises 45–52, graph two periods of each function. $$ y=\left|\cot \frac{1}{2} x\right| $$
Step-by-Step Solution
Verified Answer
In general, the graph starts at \(+\infty\), goes to \(0\) at \(\pi\), then goes back to \(+\infty\) at \(2\pi\), and it is undefined at the multiples of \(2\pi\). The pattern will repeat every \(2\pi\) because it's a periodic function.
1Step 1: Sketch the basic cotangent function
The cotangent function, \(cot(x)\), starts at \(-(infinity)\), goes up to \(0\) at \(\pi/2\) and goes to \(+infinity\) at \(pi\). It is undefined at \(0\) and at \(pi\).
2Step 2: Sketch the scaled cotangent function
Since we have cot(\(\frac{1}{2}x)\), the period of the function expands to \(2\pi\) instead of the regular \(\pi\). Hence, the graph will dip to \(0\) at \(\pi\) and \(\infty\) at \(0\) and \(2\pi\), it is undefined at the multiples of \(2\pi\).
3Step 3: Apply the absolute value
Applying the absolute value to the function will reflect all of the negative values of the function to positive. This will remove all the parts of our function that are underneath the x-axis making them appear above it.
Key Concepts
Cotangent FunctionPeriod of Trigonometric FunctionAbsolute Value Transformations
Cotangent Function
The cotangent function is one of the six fundamental trigonometric functions and is the reciprocal of the tangent function. It is represented as \(\text{cot}(x) = \frac{\text{cos}(x)}{\text{sin}(x)}\), where \(\text{sin}(x)\) and \(\text{cos}(x)\) are the sine and cosine functions, respectively.
The graph of the basic cotangent function shows a series of repeating waveforms, known as 'periods,' which have distinct characteristics. For instance, \(\text{cot}(x)\) is undefined whenever the sine function equals zero - this occurs at integer multiples of \(\text{pi}\) radians, where we notice vertical asymptotes in the graph. The function approaches infinity as it nears these asymptotes from one side and negative infinity from the other.
The typical cotangent curve has a pattern between each of its vertical asymptotes: it decreases from positive infinity, crosses the horizontal axis (where cotangent is zero), and then drops off to negative infinity as it approaches the next asymptote. In contrast to sine and cosine functions, the cotangent function does not have a maximum or minimum value but instead has these characteristic asymptotic behaviors.
The graph of the basic cotangent function shows a series of repeating waveforms, known as 'periods,' which have distinct characteristics. For instance, \(\text{cot}(x)\) is undefined whenever the sine function equals zero - this occurs at integer multiples of \(\text{pi}\) radians, where we notice vertical asymptotes in the graph. The function approaches infinity as it nears these asymptotes from one side and negative infinity from the other.
The typical cotangent curve has a pattern between each of its vertical asymptotes: it decreases from positive infinity, crosses the horizontal axis (where cotangent is zero), and then drops off to negative infinity as it approaches the next asymptote. In contrast to sine and cosine functions, the cotangent function does not have a maximum or minimum value but instead has these characteristic asymptotic behaviors.
Period of Trigonometric Function
The period of a trigonometric function refers to the horizontal length of one complete cycle of the waveform. For the basic cotangent function, this is \(\text{pi}\) radians. However, when the argument of the cotangent function is scaled, as in \(\text{cot}(\frac{1}{2}x)\), the period changes accordingly. The period of a scaled cotangent function can be found using the formula \(P = \frac{\text{pi}}{|k|}\), where \(k\) is the scaling factor.
In the example of \(y = \text{cot}(\frac{1}{2}x)\), the scaling factor is \(\frac{1}{2}\), making the period equal to \(2\text{pi}\), twice the period of the basic cotangent function. As a result, the curve extends horizontally, dipping down to zero at \(\text{pi}\) and up to infinity at multiples of \(0\) and \(2\text{pi}\). It's crucial for students to recognize how these transformations affect the graph's periodicity, as it is a common source of error.
In the example of \(y = \text{cot}(\frac{1}{2}x)\), the scaling factor is \(\frac{1}{2}\), making the period equal to \(2\text{pi}\), twice the period of the basic cotangent function. As a result, the curve extends horizontally, dipping down to zero at \(\text{pi}\) and up to infinity at multiples of \(0\) and \(2\text{pi}\). It's crucial for students to recognize how these transformations affect the graph's periodicity, as it is a common source of error.
Absolute Value Transformations
Absolute value transformations in trigonometric functions introduce a reflection across the horizontal axis for all negative values of the function. If we apply an absolute value to a trigonometric function like the cotangent, every segment of the graph that was below the horizontal axis will be reflected upwards.
This transformation completely changes the appearance of the original function. In the case of \(y = |\text{cot}(\frac{1}{2}x)|\), after the transformation, all the negative values become positive. This means all portions of the graph that were dipping down into negative values are now above the x-axis, creating a graph that consists of repeating u-shaped curves.
Furthermore, understanding absolute value transformations helps to predict the resulting graph without plotting individual points. It's also important to note that while absolute value transformations affect the positive or negative 'direction' of the graph, they don't alter the period; the period remains determined by the function's argument, as described in the previous section.
This transformation completely changes the appearance of the original function. In the case of \(y = |\text{cot}(\frac{1}{2}x)|\), after the transformation, all the negative values become positive. This means all portions of the graph that were dipping down into negative values are now above the x-axis, creating a graph that consists of repeating u-shaped curves.
Furthermore, understanding absolute value transformations helps to predict the resulting graph without plotting individual points. It's also important to note that while absolute value transformations affect the positive or negative 'direction' of the graph, they don't alter the period; the period remains determined by the function's argument, as described in the previous section.
Other exercises in this chapter
Problem 51
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