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Chapter 5: Problem 103

The graph of \(y=\cos (x)\) is shifted \(\pi\) units to the left, reflected in the \(x\) -axis, and then shifted 2 units upward. What is the equation of the curve in its final position?

Short Answer

Expert verified
The equation is \(y = \cos(x) + 2\).

Step by step solution

01

Horizontal Shift

The graph of the function is shifted \(\pi\) units to the left. To achieve this, we replace \(x\) with \(x + \pi\) in the cosine function. Thus, the function becomes \[ y = \cos(x + \pi) \]
02

Reflection in the x-Axis

Next, we reflect the function in the \(x\)-axis. Reflecting across the \(x\)-axis involves multiplying the whole function by \(-1\). Therefore, the equation becomes \[ y = -\cos(x + \pi) \]
03

Simplify the Cosine Shift

Use the cosine function's periodic properties to simplify further. Note that \(\cos(x + \pi) = -\cos(x)\). Substituting this into the equation, we get \[y = -(-\cos(x)) = \cos(x)\]
04

Vertical Shift

Finally, shift the function upward by 2 units. This is done by adding 2 to the function. So the final equation is \[ y = \cos(x) + 2\]

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Key Concepts

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

Understanding the Cosine Function
The cosine function, represented as \(y = \cos(x)\), is one of the fundamental trigonometric functions. The graph of \cos(x)\ resembles a wave that oscillates between \1\ and \ -1 \. It demonstrates a repeating pattern every \2\蟺 units along the x-axis. This oscillatory nature is due to the periodic property of the cosine function. Key features of this function include:
  • The maximum value is 1, occurring at \x = 0, 2\蟺, 4\蟺, and so on.
  • The minimum value is -1, occurring at \x = \pi, 3\蟺, 5\蟺, and so on.
  • It crosses the x-axis at \x = \pi\2, 3\蟺\2, 5\蟺\2, etc.
Understanding these properties helps us analyze transformations applied to the cosine function effectively.
Horizontal Shift
A horizontal shift involves moving the graph of a function left or right along the x-axis. To shift the cosine function \pi\ units to the left, each x value in the original function is replaced with \x + \pi\. For the standard cosine function \(y = \cos(x)\), this results in the function \(y = \cos(x + \pi)\). This transformation adjusts the starting point of the wave while retaining its shape. As a result, all key points from the graph of \cos(x)\ are shifted to the left by \pi\ units.
Reflection
Reflection is a transformation that flips the graph over a specific axis. Reflecting the cosine function \(y = \cos(x)\) across the x-axis involves multiplying the entire function by \ -1\. The reflected function becomes \(y = \ -\cos(x)\). This transformation converts each positive value of \cos(x)\ to a negative value, and vice versa. When applying this to our horizontally shifted function \(y = \cos(x + \pi)\), we get \(y = \ -\cos(x + \pi)\). It鈥檚 essential to remember that reflecting a function can significantly alter its visual representation by inverting its peaks and valleys.
Vertical Shift
A vertical shift translates the entire graph of a function up or down along the y-axis. For the final step in this problem, we shift the function \cos(x)\ upward by 2 units. Mathematically, this is done by adding 2 to the function, resulting in \(y = \cos(x) + 2\). A vertical shift impacts all y-values, elevating the whole graph while preserving its overall shape and orientation. After shifting, the function鈥檚 minimum value moves from \ -1 \ to 1, and its maximum value moves from 1 to 3.
This transformation completes the process, bringing us to the final function: \(y = \cos(x) + 2 \).

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Most popular questions from this chapter

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