Problem 16 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The graph of y = cos(x - pi) is the same graph as y = cos(x), but shifted to the right by \(\pi\) units. Thus, in the interval from 0 to 2\(\pi\), the graph will start from x=\(\pi/2\) at 0, and end at x = \((3/2)\pi\) at 0, having descended to -1 at x = \(\pi\).

Step 1: Understand the function to be graphed

We are asked to graph the function \(y=\cos(x - \pi)\). Here, the value of \(-\pi\) is subtracted from \(x\) inside the cosine function. This corresponds to a phase shift. The phase shift happens horizontally - a positive value would shift the graph to the right, and a negative value would shift the graph to the left. In this case, the function \(y=\cos(x - \pi)\) will be shifted to the right by \(\pi\) units.

Step 2: Draw the original cosine graph

First, let's start out by drawing the original graph of \(y=\cos(x)\) without the phase shift. For cos(x), one period starts at x=0 and ends at \(x = 2\pi\). The maximum value is 1 (at x=0) and the minimum value is -1 (at \(x=\pi\)). Cosine function returns to maximum value at \(x = 2\pi\). This information is used to make a rough sketch of the cosine wave.

Step 3: Apply the phase shift

To apply the phase shift, we go to every point on the original \(y=\cos(x)\) graph and move it right by \(\pi\) units. This means that the peak of the wave, which was originally at \(x = 0\), now occurs at \(x = \pi\), and the minimum point, which was originally at \(x = \pi\), now occurs at \(x = 2\pi\). The wave now completes one period from \(x = \pi\) to \(x = 3\pi\). Since we are only considering from \(0\) to \(2\pi\), the graph will begin at x=\(\pi/2\) with a value of 0, descend to -1 at x = \(\pi\), and return to 0 at x = \((3/2)\pi\).

Step 4: Label the graph

Finally, we should annotate the graph with important points and labels. Key points to label would include the \(\pi\) shift, the maximum at (1, \(\pi\)), and the minimum at (-1, \(2\pi\)).

Key Concepts

Phase Shift in Trigonometry

Phase shift is a crucial concept in trigonometry that describes how a function is shifted horizontally on the graph. When you look at the trigonometric function like cosine, the phase shift tells you how much the graph is moved left or right along the x-axis. This is governed by the expression inside the function, specifically any addition or subtraction that occurs there.

For instance, in the expression \(y=\cos(x-\pi)\), the \(-\pi\) tells us to shift the entire cosine wave to the right by \(\pi\) units. Think of it like adjusting a slider; the negative sign indicates a shift to the right, while positive would shift it left. This shift effectively changes the starting point of the function's cycle. Since cosine is periodic with a period of \(2\pi\), understanding where this shift puts the function in its cycle is vital for graphing accurately.

To visualize it, if you were watching a movie, a phase shift to the right would mean you're starting from a later part of the movie, skipping the original starting scenes.

Cosine Function Graph

The cosine function, \(y=\cos(x)\), is one of the fundamental trigonometric functions that produces a wave-like graph. The standard cosine function is periodic with a period of \(2\pi\), meaning its pattern repeats every \(2\pi\) units along the x-axis.

This graph starts at its maximum value of 1 when \(x=0\), decreases to a minimum at \(-1\) when \(x=\pi\), and comes back to its maximum at \(x=2\pi\).

• The wave begins at its highest point.
• The middle of the wave touches the x-axis (value of 0).
• The lowest point occurs halfway through the period.

These are key markers on the graph and understanding them helps in plotting the graph quickly and accurately. It is this predictable pattern that makes cosine graphs so useful in various applications, from physics to engineering. When plotting the graph, these main points serve as a good frame of reference to draw the smooth wave that typifies the cosine function.

Trigonometric Function Transformations

Transformations of trigonometric functions involve altering the basic sine or cosine graphs in terms of their amplitude, period, phase shift, or vertical shift. These transformations allow the graph to stretch, shrink, shift, or mirror.

• **Amplitude Change:** This affects the height of the wave. For example, multiplying the cosine function by 2 (\(y=2\cos(x)\)) doubles its peaks and troughs.
• **Period Change:** This alters the distance required for the function to repeat itself. For example, \(y=\cos(2x)\) halves the period to \(\pi\).
• **Phase Shift:** Discussed earlier, this moves the graph horizontally along the x-axis.
• **Vertical Shift:** This moves the entire function up or down on the graph. For instance, \(y=\cos(x) + 1\) lifts the wave one unit higher.

Each transformation is a modification of the function's original parameters, changing the way the wave looks on the graph. When graphing trigonometric functions that include transformations, it's essential to consider each change in the proper order. This ensures you're accurately plotting the new function based on its alterations from the standard form.