Problem 14 - Algebra 2

Question

Graph each translation of \(y=\cos x\) in the interval from 0 to 2\(\pi\) $$ y=\cos (x-4) $$

Step-by-Step Solution

Answer

The graph of y = cos(x - 4) is a standard cosine function graph, shifted 4 units to the right.

Step 1: Understanding the standard cosine function graph

First, graph the standard cosine function in the interval from 0 to 2*pi. The standard cos(x) function forms a wave-like graph and repeats every 2*pi*. The graph starts at (0,1), goes down to (pi, -1), and comes back up to (2*pi, 1). It is important to keep reference points in mind when trying to graph a transformation.

Step 2: Apply the phase shift to the cosine function

The provided equation is y=cos(x-4). This means that for any given x in the cos(x) function, you'll be substituting (x - 4). This changes the x-coordinates of all points in the original cos(x). Each point is shifted 4 units to the right.

Step 3: Graphing the translated cosine function

Start plotting your function, again, using the standard shape of the cosine function for reference. Because of the phase shift, the wave will start at (4,1) instead of at (0,1). The bottom of the wave will now be located at (4+pi, -1), and it will come back up to complete a wave at (4+2*pi, 1). Repeat this process until you have graphed all points in the given interval.

Key Concepts

Cosine Function

The cosine function, denoted as \(y = \cos x\), is one of the fundamental trigonometric functions. It represents the x-coordinate of a point on the unit circle as it moves around, exhibiting wave-like behavior. The graph of the cosine function is a smooth, continuous wave that repeats its shape every \(2\pi\), which is its period.

Key aspects of the cosine graph include:

• The maximum value: 1, occurs at \(x = 0, 2\pi, 4\pi, \ldots\)
• The minimum value: -1, happens at \(x = \pi, 3\pi, 5\pi, \ldots\)
• Zero crossings: the graph passes through the x-axis at \(x = \pi/2, 3\pi/2, 5\pi/2, \ldots\)

The cosine function starts at its highest point and then dips down to its lowest point before returning back up. This distinctive pattern, with its crest at the start, makes cosine particularly useful for modeling waves and oscillations in various fields of science and engineering.

Phase Shift

A phase shift involves shifting the function horizontally along the x-axis. The given equation \(y = \cos(x - 4)\) demonstrates such a phase shift.

This phase shift affects the function by translating all its points to the right by 4 units. In simple terms, each point on the standard \(\cos x\) graph moves from \(x\) to \(x + 4\).

To visualize it:

• The starting point originally at \((0, 1)\) moves to \((4, 1)\).
• The point at \((\pi, -1)\) shifts to \((\pi + 4, -1)\).
• The end point for one period at \((2\pi, 1)\) moves to \((2\pi + 4, 1)\).

This shift does not alter the shape or size of the graph—it merely slides it sideways without changing its amplitude, frequency, or period. Thus, it is particularly useful in adjusting functions to fit the desired phase alignment in applications.

Interval [0, 2π]

The interval \([0, 2\pi]\) is a critical domain for understanding the behavior of trigonometric functions like \(\cos x\). This interval represents the full cycle of the cosine wave, from its start through one complete period, returning to its original position.

Within this interval:

• The graph begins at \(x = 0\) and ends exactly one period later at \(x = 2\pi\).
• Important points, such as peaks and troughs, are clearly marked.
• It allows for analyzing how transformations affect the wave, since it encompasses one full oscillation.

For the modified function \(y = \cos(x - 4)\), we observe its behavior in this same interval by adjusting our x-values accordingly. Even if we apply transformations like phase shifts, considering their influence within the interval \([0, 2\pi]\) ensures we understand the function's impact over a complete cycle.