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Chapter 5: Problem 92

Write the equation of each curve in its final position. The graph of \(y=\cot (x)\) is shifted \(\pi / 2\) units to the left, reflected in the \(x\) -axis, then translated 1 unit upward.

Short Answer

Expert verified
y = -cot(x + \pi / 2) + 1

Step by step solution

01

- Shift \( y = \cot(x) \) \( \pi / 2\) units to the left

To shift the graph to the left by \( \pi / 2\) units, replace \( x \) with \( x + \pi / 2 \) in the function: \( y = \cot(x + \pi / 2) \).
02

- Reflect over the x-axis

To reflect the graph over the x-axis, multiply the entire function by \( -1 \): \( y = -\cot(x + \pi / 2) \).
03

- Translate 1 unit upward

To translate the graph 1 unit upward, add 1 to the entire function: \( y = -\cot(x + \pi / 2) + 1 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cotangent Function
The cotangent function, denoted as \(\text{cot}(x)\), is one of the primary trigonometric functions. It is defined as the reciprocal of the tangent function, so \(\text{cot}(x) = \frac{1}{\text{tan}(x)}\). This function is periodic with a period of \(\text{蟺}\). It has vertical asymptotes where the tangent function is zero, which occurs at integer multiples of \(\text{蟺}\). The cotangent function graph looks similar to a series of decreasing curves spaced \(\text{蟺}\) units apart. Understanding the base graph of \(\text{cot}(x)\) is essential when performing various transformations on it.
Horizontal Shift
A horizontal shift involves moving the graph of a function left or right along the x-axis. For the cotangent function, shifting the graph to the left by \(\frac{蟺}{2}\) units means each point on the graph moves \(\frac{蟺}{2}\) units to the left. Mathematically, this is done by replacing \(x\) in the function with \(x + \frac{蟺}{2}\). So, if we start with \(y=\text{cot}(x)\), shifting it left by \(\frac{蟺}{2}\) units modifies it to \(y=\text{cot}(x + \frac{蟺}{2})\). This transformation changes the position of the vertical asymptotes and the shape of the curves but retains the overall periodic nature.
Reflection
Reflecting a graph over the x-axis involves flipping it upside down. This is achieved by multiplying the entire function by \(-1\). For the cotangent function, the equation \(y=\text{cot}(x + \frac{蟺}{2})\) becomes \(y=-\text{cot}(x + \frac{蟺}{2})\) after reflection. This transformation turns positive y-values into negative ones and vice versa. Reflections over the x-axis can help in understanding symmetric properties and inverse relationships in trigonometric functions.
Vertical Translation
A vertical translation moves the graph of a function up or down along the y-axis. To translate the graph of the cotangent function upward by 1 unit, we simply add 1 to the function. So after shifting left and reflecting, we start with \(y=-\text{cot}(x + \frac{蟺}{2})\). Adding 1 to this function results in \(y=-\text{cot}(x + \frac{蟺}{2}) + 1\). This ensures every point on the graph moves up by one unit. Vertical translations help in adjusting the position of the graph to better fit a specific situation or context.

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Most popular questions from this chapter

Area of a Sector of a Circle If a slice with central angle \(\alpha\) radians is cut from a pizza of radius \(r,\) then what is the area of the slice?

Convert each angle to decimal degrees. When necessary, round to four decimal places. $$-5^{\circ} 45^{\prime} 30^{\prime \prime}$$

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