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b) y = – cot (x) + 2

b) y = – cot (x) + 2

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Analysis of the Function \( y = -\cot\left(\frac{\pi}{2} x\right) + 2 \) (Degrees) **Function Parameters:** - **Flip:** - The function has a vertical flip because of the negative sign in front of the cotangent function. - **Vertical Shift:** - The function is shifted upwards by 2 units. - **Period:** - The basic period of the cotangent function \( \cot(x) \) is \( 180^\circ \) (or \( \pi \) radians). - Since the argument of the cotangent function is \( \frac{\pi}{2} x \), the period is calculated as follows: \\ Period = \( \frac{180^\circ}{\frac{\pi}{2}} = \frac{180^\circ \cdot 2}{\pi} = 360^\circ \). - **Phase Shift:** - There is no phase shift in this function as there are no horizontal shifts in the argument of the function. - **Domain:** - The cotangent function is undefined where its argument is an integer multiple of \( 180^\circ \) (or \(\pi\)). - Therefore, the domain is: \\ \( x \neq \frac{180^\circ k}{\frac{\pi}{2}} = 360^\circ k \), where \( k \) is an integer. - **Range:** - The range of the cotangent function spans all real numbers, but due to the vertical shift upwards by 2 units, the range also spans all real numbers: \( (-\infty, \infty) \). **Graph Explanation:** - The graph shows both the x-axis (in degrees, marked at intervals of 60 degrees ranging from -360º to 360º) and the y-axis (with a unit scale). - There's a notable characteristic of the cotangent function, which tends to have vertical asymptotes at intervals of the period, which are at \( x = 360^\circ k \) due to the adjustment in the argument. - Evaluated points of interest would include the critical behavior where the function tends toward either positive or negative infinity at these asymptotes. - In general, the cotangent function decreases between vertical asymptotes before it repeats its value every 360º interval. By understanding
Transcribed Image Text:### Analysis of the Function \( y = -\cot\left(\frac{\pi}{2} x\right) + 2 \) (Degrees) **Function Parameters:** - **Flip:** - The function has a vertical flip because of the negative sign in front of the cotangent function. - **Vertical Shift:** - The function is shifted upwards by 2 units. - **Period:** - The basic period of the cotangent function \( \cot(x) \) is \( 180^\circ \) (or \( \pi \) radians). - Since the argument of the cotangent function is \( \frac{\pi}{2} x \), the period is calculated as follows: \\ Period = \( \frac{180^\circ}{\frac{\pi}{2}} = \frac{180^\circ \cdot 2}{\pi} = 360^\circ \). - **Phase Shift:** - There is no phase shift in this function as there are no horizontal shifts in the argument of the function. - **Domain:** - The cotangent function is undefined where its argument is an integer multiple of \( 180^\circ \) (or \(\pi\)). - Therefore, the domain is: \\ \( x \neq \frac{180^\circ k}{\frac{\pi}{2}} = 360^\circ k \), where \( k \) is an integer. - **Range:** - The range of the cotangent function spans all real numbers, but due to the vertical shift upwards by 2 units, the range also spans all real numbers: \( (-\infty, \infty) \). **Graph Explanation:** - The graph shows both the x-axis (in degrees, marked at intervals of 60 degrees ranging from -360º to 360º) and the y-axis (with a unit scale). - There's a notable characteristic of the cotangent function, which tends to have vertical asymptotes at intervals of the period, which are at \( x = 360^\circ k \) due to the adjustment in the argument. - Evaluated points of interest would include the critical behavior where the function tends toward either positive or negative infinity at these asymptotes. - In general, the cotangent function decreases between vertical asymptotes before it repeats its value every 360º interval. By understanding
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