In Exercises 17–24, graph two periods of the given cotangent function. y = 3 cot(x + π/2)

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Recall the general form of the cotangent function: \(y = A \cot(Bx + C)\), where \(A\) affects the amplitude (vertical stretch), \(B\) affects the period, and \(C\) is the phase shift.

Identify the parameters in the given function \(y = 3 \cot(x + \frac{\pi}{2})\): here, \(A = 3\), \(B = 1\), and \(C = \frac{\pi}{2}\).

Calculate the period of the cotangent function using the formula \(\text{Period} = \frac{\pi}{|B|}\). Since \(B = 1\), the period is \(\pi\).

Determine the phase shift by solving \(Bx + C = 0\), which gives \(x = -\frac{C}{B} = -\frac{\pi}{2}\). This means the graph is shifted to the left by \(\frac{\pi}{2}\).

To graph two periods, plot the cotangent function starting from \(x = -\frac{\pi}{2}\) and extend the graph over an interval of length \(2 \times \pi = 2\pi\). Mark key points such as zeros, vertical asymptotes, and the shape of the curve scaled by the amplitude 3.

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주요 개념

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Cotangent Function and Its Properties

The cotangent function, cot(x), is the reciprocal of the tangent function and is defined as cos(x)/sin(x). It has vertical asymptotes where sin(x) = 0, i.e., at integer multiples of π, and its period is π. Understanding its shape and behavior is essential for graphing.

Phase Shift in Trigonometric Functions

A phase shift occurs when the input variable x is replaced by (x + c), shifting the graph horizontally. For y = cot(x + π/2), the graph shifts left by π/2 units. Recognizing this shift helps in correctly positioning the graph on the x-axis.

Amplitude and Vertical Stretch

The coefficient 3 in y = 3 cot(x + π/2) vertically stretches the cotangent graph by a factor of 3. While cotangent has no maximum or minimum values, this stretch affects the steepness of the curve, making it rise and fall more sharply.