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Chapter 13: Problem 18

Describe any phase shift and vertical shift in the graph. $$ y=4 \cos (x+1)-2 $$

Short Answer

Expert verified
The phase shift is -1 unit to the left and the vertical shift is -2 units downward.

Step by step solution

01

Identify the phase shift

In the given function \(y=4\cos(x+1)-2\), the phase shift (h) can be found inside the cosine function and it is -1. Therefore, the phase shift is -1 unit to the left.
02

Identify the vertical shift

The vertical shift (k) can be found outside the cosine functions, which is -2 in this case. Thus, the graph of the given function shifts down by 2 units.
03

Finalize the Result

By combining the result from step 1 and step 2, we can conclude that the graph of the function \(y=4\cos(x+1)-2\) has a phase shift of -1 unit to the left and a vertical shift of -2 units downwards.

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Key Concepts

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

Vertical Shift
A vertical shift in a function occurs when a constant is added or subtracted from the entire function. This addition or subtraction causes the graph of the function to move up or down. In the context of trigonometric functions, such as the cosine or sine functions, this shift is usually represented as a constant term.
For example, in the function given in the exercise, \[ y = 4\cos(x+1)-2 \]the constant -2 represents a vertical shift. Since -2 is subtracted, the graph shifts downward by 2 units. This means every point on the graph of the original cosine function is moved two units lower on the y-axis.
Graph of Trigonometric Functions
The graphs of trigonometric functions show periodic behavior, characterized by their repeating wave-like patterns. These functions include sine, cosine, tangent, and their associated reciprocal functions.
The cosine function, in particular, typically starts at a maximum point when x is zero, before it waves down, reaches a minimum, and comes back up to the maximum in 360 degrees (or \(2\pi\) radians).
Knowing the inherent shape of these graphs helps us understand how values such as the phase shift and vertical shift can change their appearance on a graph.
Cosine Function
The cosine function is a fundamental trigonometric function, often written as \( \cos(x) \). This function relates to the x-coordinate of a point on the unit circle corresponding to a given angle.
The basic graph of the cosine function is a wave that oscillates between 1 and -1, with a period of \(2\pi \). Each complete wave represents one full cycle of the cosine function. This means the function repeats its values every \(2\pi \) units as it moves along the x-axis.
When transformations are applied, such as amplitude changes or phase shifts, these affect the cosine wave by modifying its height, horizontal positioning, or other fundamental aspects.
Transformations of Functions
Transformations of functions involve changing the appearance of a function's graph. Several types of transformations can be applied to functions, including trigonometric functions like sine and cosine. These transformations help to model various real-world situations or analyze behavior changes.
Some common types of transformations include:
  • Phase Shift: Adjusts the horizontal position of the function by shifting the graph left or right.
  • Vertical Shift: Moves the graph up or down by adding or subtracting a constant.
  • Amplitude Change: Increases or decreases the height of the wave, affecting how high or low the graph oscillates.
  • Reflection: Flips the graph across a given axis, which can change the direction of the waves.
Understanding these transformations allows us to accurately sketch the graph or interpret modifications of trigonometric functions.

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