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

Verified step by step guidance

1. Identify the general form of the cotangent function: \(y = A \cot(Bx)\), where \(A\) is the amplitude (vertical stretch) and \(B\) affects the period of the function.
2. Determine the period of the cotangent function using the formula \(\text{Period} = \frac{\pi}{|B|}\). Here, \(B = \frac{\pi}{2}\), so calculate the period as \(\frac{\pi}{\frac{\pi}{2}}\).
3. Calculate two full periods by multiplying the period by 2, which will give the length along the x-axis to graph.
4. Identify key points within one period of the cotangent function, such as where the function is undefined (vertical asymptotes) and where it crosses the x-axis, then replicate these points for the second period.
5. Use the amplitude \(A = -3\) to stretch and reflect the cotangent curve vertically, then sketch the graph over the two periods determined, including vertical asymptotes and the shape of the curve between them.

Key Concepts

Cotangent Function and Its Graph

The cotangent function, cot(x), is the reciprocal of the tangent function and has vertical asymptotes where sine is zero. Its graph consists of repeating curves between these asymptotes, with zeros where cosine is zero. Understanding its shape and behavior is essential for accurate graphing.

Period of the Cotangent Function

The period of cotangent is normally π, but when the function is y = cot(bx), the period changes to π/|b|. For y = -3 cot(π/2 x), the period is 2 because π divided by (π/2) equals 2. Knowing the period helps in determining the length of one full cycle on the x-axis.

Amplitude and Vertical Stretch

Although cotangent does not have a maximum or minimum amplitude like sine or cosine, the coefficient outside the function, here -3, vertically stretches and reflects the graph. The negative sign flips the graph over the x-axis, and the factor 3 stretches it, affecting the steepness of the curves.