Question

Determine the period (in degrees) of each function. Then, use the language of transformations to describe how each graph is related to the graph of \(y=\cos x\) a) \(y=\cos 2 x\) b) \(y=\cos (-3 x)\) c) \(y=\cos \frac{1}{4} x\) d) \(y=\cos \frac{2}{3} x\)

Short answer

a) Period: 180 degrees, horizontal compression by 2. b) Period: 120 degrees, horizontal compression by 3. c) Period: 1440 degrees, horizontal stretch by 4. d) Period: 540 degrees, horizontal stretch by \(\frac{3}{2}\).

Step-by-step solution

Understand the periodicity of the cosine function

The standard cosine function, given by \(y = \cos x\), has a period of 360 degrees. This means that the function repeats its values every 360 degrees.

Determine the period formula for transformed cosine functions

For functions of the form \(y = \cos (k x)\), the period can be found using the formula \(\text{Period} = \frac{360}{|k|}\). Here, \(k\) is the coefficient of \(x\).

Calculate the period for \(y = \cos 2 x\)

Using the formula \(\text{Period} = \frac{360}{|k|}\) with \(k = 2\), we get \(\text{Period} = \frac{360}{2} = 180\) degrees. Hence, the period of \(y = \cos 2 x\) is 180 degrees.

Describe the transformation for \(y = \cos 2 x\)

The function \(y = \cos 2 x\) is a horizontal compression by a factor of 2 of the graph of \(y = \cos x\).

Calculate the period for \(y = \cos (-3 x)\)

Using the formula \(\text{Period} = \frac{360}{|k|}\) with \(k = -3\), we get \(\text{Period} = \frac{360}{3} = 120\) degrees. Hence, the period of \(y = \cos (-3 x)\) is 120 degrees.

Describe the transformation for \(y = \cos (-3 x)\)

The function \(y = \cos (-3 x)\) is a horizontal compression by a factor of 3 of the graph of \(y = \cos x\). The negative sign only indicates a reflection over the y-axis, but for cosine, this reflection does not change the graph.

Calculate the period for \(y = \cos \frac{1}{4} x\)

Using the formula \(\text{Period} = \frac{360}{|k|}\) with \(k = \frac{1}{4}\), we get \(\text{Period} = \frac{360}{\frac{1}{4}} = 1440\) degrees. Hence, the period of \(y = \cos \frac{1}{4} x\) is 1440 degrees.

Describe the transformation for \(y = \cos \frac{1}{4} x\)

The function \(y = \cos \frac{1}{4} x\) is a horizontal stretch by a factor of 4 of the graph of \(y = \cos x\).

Calculate the period for \(y = \cos \frac{2}{3} x\)

Using the formula \(\text{Period} = \frac{360}{|k|}\) with \(k = \frac{2}{3}\), we get \(\text{Period} = \frac{360}{\frac{2}{3}} = 540\) degrees. Hence, the period of \(y = \cos \frac{2}{3} x\) is 540 degrees.

Describe the transformation for \(y = \cos \frac{2}{3} x\)

The function \(y = \cos \frac{2}{3} x\) is a horizontal stretch by a factor of \(\frac{3}{2}\) of the graph of \(y = \cos x\).

Key concepts

Periodicity of Trigonometric Functions

Understanding the concept of periodicity is central to trigonometry. For trigonometric functions like the cosine function, periodicity refers to the characteristic that the function repeats its values at regular intervals. For instance, the standard cosine function, represented as \(y = \cos x\), has a period of 360 degrees. This means that every 360 degrees, the function values repeat. When we modify this function, such as \(y = \cos (kx)\), the period changes. The formula to determine the new period is \( \text{Period} = \frac{360}{|k|}\), where \(k\) is the coefficient of \(x\). Applying this formula helps in understanding how the graph of the function is altered.

Horizontal Compression

Horizontal compression is a key transformation for trigonometric functions. It occurs when we multiply the variable \(x\) by a constant greater than 1. For example, in the function \(y = \cos (2x)\), the 2 in the argument of the cosine function causes the graph to compress horizontally by a factor of 2. To see the effect on the period, use the formula \( \text{Period} = \frac{360}{|k|} \). For \(y = \cos (2x)\), we get \( \text{Period} = \frac{360}{2} = 180 \) degrees. Hence, the graph repeats every 180 degrees, instead of the original 360 degrees. This means the graph will look 'squeezed' together when compared to the standard cosine function.

Horizontal Stretch

Horizontal stretch is another transformative concept for trigonometric functions. This occurs when we multiply the variable \(x\) by a constant between 0 and 1. Consider the function \(y = \cos (\frac{1}{4} x)\). The \(\frac{1}{4}\) causes the graph to stretch horizontally by a factor of 4. Using the period formula, \( \text{Period} = \frac{360}{|k|}\), we find that \(y = \cos (\frac{1}{4} x)\) has a period of \( \text{Period} = \frac{360}{\frac{1}{4}} = 1440 \) degrees. Hence, the graph stretches out, repeating every 1440 degrees. This transformation makes the graph appear 'stretched' out compared to the original cosine function.

Trigonometric Transformations

Trigonometric transformations involve various modifications to trigonometric functions, affecting their amplitude, period, phase shift, and vertical shift. Two important transformations are horizontal compression and stretch:
- Horizontal Compression: When \(k\) is greater than 1, as seen in \(y = \cos (2x)\), the function compresses horizontally.
- Horizontal Stretch: When \(k\) is between 0 and 1, as in \(y = \cos (\frac{1}{4} x)\), the function stretches horizontally.
These transformations can be analyzed by the period formula, \( \text{Period} = \frac{360}{| k |}\), making it straightforward to understand and predict the behavior of the transformed function. For example, \(y = \cos (-3 x)\) involves a horizontal compression by 3 and has a period of 120 degrees, while the negative sign indicates a reflection, which doesn't change the graph for cosine functions.