Problem 92 - Calculus for Biology and Medicine

Question

For each given pair of functions, use a graphing calculator to compare the functions. Describe what you see. \(y=\cos x\) and \(y=\cos (x+1)\)

Step-by-Step Solution

Answer

\(y = \cos(x+1)\) is \(y = \cos x\) shifted 1 unit left.

Step 1: Understanding the Functions

We have two functions, \(y = \cos x\) and \(y = \cos(x+1)\). The first function is the standard cosine function, and the second is a cosine function with a horizontal shift. The task is to analyze how these functions differ when graphed.

Step 2: Graphing the Standard Cosine Function

Plot the graph of \(y = \cos x\). This function has a periodic wave shape with peaks at 1 and troughs at -1. It completes one full cycle from \(0\) to \(2\pi\) radians.

Step 3: Graphing the Shifted Cosine Function

Now plot \(y = \cos(x+1)\). This function is identical in shape to \(y = \cos x\) but horizontally shifted to the left by 1 unit. Each point on the graph of \(y = \cos(x+1)\) is directly 1 unit to the left of the corresponding point on \(y = \cos x\).

Step 4: Comparing the Graphs

Compare both graphs. You'll notice that \(y = \cos(x+1)\) starts its cycle 1 unit earlier than \(y = \cos x\). The phase shift indicates a direct horizontal movement without any changes to amplitude or period.

Step 5: Concluding the Analysis

The graph of \(y = \cos(x+1)\) is the same as \(y = \cos x\) but is shifted left by 1 unit. The amplitude and period remain unchanged at 1 and \(2\pi\) respectively.

Key Concepts

Phase Shift

Phase shift refers to the horizontal translation of a trigonometric graph to the left or right. In the case of the cosine function you have, this concept is demonstrated by the shift from \(y = \cos x\) to \(y = \cos(x+1)\). The addition inside the parentheses results in shifting the graph horizontally. Specifically, \(y = \cos(x+1)\) translates the graph of \(y = \cos x\) one unit to the left.

A simple way to determine the direction and magnitude of a phase shift is by looking at the expression inside the cosine function:

• If the function is \(y = \cos(x + C)\), the graph shifts \(C\) units to the left.
• If the function is \(y = \cos(x - C)\), the graph shifts \(C\) units to the right.

It's important to note that phase shifts do not affect the amplitude (the height of the peaks and troughs) or the period (the length of one full cycle) of the function. They simply translate the entire graph across the x-axis.

Trigonometric Graphs

Trigonometric graphs, such as those of sine and cosine, show the relationship between an angle and ratio of sides in a right triangle. These graphs are periodic, meaning they repeat their shape after a certain interval known as the period. For cosine, the graph shape is typically a smooth wave that rises and falls between 1 and -1, repeating every \(2\pi\) radians.

The key characteristics of trigonometric graphs include:

• Amplitude: It determines the height of the wave, which is the distance from the middle of the wave (axis) to the peak. For the cosine function \(y = \cos x\), the amplitude is 1.
• Period: The distance required for the function to complete one full cycle. For cosine, this is \(2\pi\).
• Phase Shift: As mentioned, it indicates how much the graph has moved left or right.
• Vertical Shift: Determines whether the whole graph moves up or down along the y-axis - not applicable in this set of functions as it is 0.

Studying these characteristics is crucial for predicting how changes in a trigonometric equation affect its graph.

Graphing Calculator

A graphing calculator is a vital tool when investigating the visual representation of complex functions like trigonometric graphs. By inputting different functions, students can see firsthand how changes in an equation affect its shape and position on the coordinate plane.

How to use a graphing calculator effectively:

• Enter the given functions into the calculator to plot the graphs. For example, input \(y = \cos x\) and \(y = \cos(x+1)\) separately.
• Use the zoom function to adjust the view so that you're able to see the important features of the graphs clearly, such as intercepts, peaks, and troughs.
• Leverage the "trace" feature to move along the graph and observe coordinates, which can be helpful to understand the phase shift in step-by-step detail.

By using the calculator, the abstract parts of mathematics become more tangible, providing a deeper understanding. Students can visualize the phase shifts and other transformations in real-time, cementing their comprehension of trigonometric graphs.