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

Verified step by step guidance

1. 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.
2. Identify the parameters in the given function \(y = 3 \cot(x + \frac{\pi}{2})\): here, \(A = 3\), \(B = 1\), and \(C = \frac{\pi}{2}\).
3. Calculate the period of the cotangent function using the formula \(\text{Period} = \frac{\pi}{|B|}\). Since \(B = 1\), the period is \(\pi\).
4. 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 \(\frac{\pi}{2}\).
5. 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.

Key Concepts

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.