Graph and answer ### 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

The flip of a function is defined when the graph of the function is reflected about the x-axis. The…

Answered: b) y = – cot (x) + 2

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b) y = – cot (x) + 2

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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Graph and answer

### 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.

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