Problem 93 - 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 \(y=2 \cos x\)

Step-by-Step Solution

Answer

The function \(y = 2 \cos x\) has double the amplitude of \(y = \cos x\), but the same period.

Step 1: Identify the Base Function

The base function provided is the cosine function \(y = \cos x\). The cosine function is periodic with a period of \(2\pi\) and an amplitude of 1.

Step 2: Analyze the Transformation

The second function is \(y = 2 \cos x\). This function is a transformation of the base cosine function with changes in amplitude. The leading coefficient 2 multiplies the cosine function, affecting its amplitude, but the period remains \(2\pi\) since there's no change to the \(x\) coefficient.

Step 3: Graph the Base Function

Using a graphing calculator, plot the base function \(y = \cos x\). It oscillates between 1 and -1 with a period of \(2\pi\), and peaks at 1 and valleys at -1.

Step 4: Graph the Transformed Function

Plot the transformed function \(y = 2 \cos x\) on the same graph. Observe that the graph oscillates between 2 and -2. The amplitude is doubled, so the peaks reach 2 and the valleys reach -2, while the period remains unchanged.

Step 5: Compare and Describe

By comparing both graphs, note that \(y = 2 \cos x\) has the same periodicity as \(y = \cos x\), but the amplitude is increased. The transformed function stretches vertically by a factor of 2.

Key Concepts

Cosine Function

The cosine function, mathematically denoted as \(y = \cos x\), is a fundamental trigonometric function. Its graph is a smooth curve called a cosine wave. This wave is periodic, which means that it repeats itself at regular intervals. For the cosine function, these intervals occur every \(2\pi\) units along the x-axis. This period signifies one complete cycle of the function from the start to the end of the repeating pattern.

The range of the cosine function spans from -1 to 1. This is because as the angle \(x\) changes, \(\cos x\) takes values within this range. When graphed, the cosine function creates peaks at 1 and troughs at -1. The line graph starts at its maximum value of 1 when \(x = 0\) and demonstrates symmetry around both the y-axis and the origin.

• Period: \(2\pi\)
• Amplitude: 1
• Range: -1 to 1

The graph of \(y = \cos x\) looks like a wave consistently moving up and down, which forms a crucial basis for understanding transformations that lead to the modifying of its shape and range.

Graph Transformations

Graph transformations refer to modifications made to a function's graph, resulting in a change of shape, size, position, or orientation. These transformations are pivotal in graphing different variations of base functions like the cosine function. A transformation can include shifting, stretching, compressing, or reflecting a graph.

In the exercise, the cosine function \(y = \cos x\) is transformed to \(y = 2 \cos x\). The multiplier of 2 applied to \(\cos x\) illustrates a vertical stretch in the graph. This doesn't alter the function's period. Both the original and transformed functions maintain a periodicity of \(2\pi\).

Common transformations include:

• **Vertical Stretch/Compression:** Changes induced by multiplying the function by a constant (as in \(y = 2\cos x\)).
• **Horizontal Shift:** Moving the graph left or right by adjusting the x-variable.
• **Vertical Shift:** Moving the graph up or down by adding or subtracting from the function.
• **Reflection:** Flipping the graph over an axis using negative variables.

Understanding these transformations allows the manipulation of basic trigonometric graphs to represent more complex functions.

Amplitude of Trigonometric Functions

Amplitude is a crucial aspect of trigonometric functions and represents the height of the peaks of the wave from the centerline of the graph. Specifically, in the context of the cosine function, the amplitude determines how far the function's value can reach from its central axis.

For the base function \(y = \cos x\), the amplitude is 1, as the graph spans from 1 to -1. However, when vertical transformations occur, such as in the function \(y = 2 \cos x\), the amplitude changes. The coefficient '2' affects the amplitude, doubling it because the graph extends from 2 to -2.

Key takeaways on amplitude transformations:

• Amplitude is always positive and is given by the absolute value of the coefficient before the trigonometric term.
• It directly modifies the height of wave peaks and troughs from the graph's center line.
• The period remains unaffected by changes in amplitude.

By understanding amplitude, one can predict the extent of a trigonometric function's variation and therefore graph it accurately, taking into account any transformations applied.