Question

Graph each function using transformations or the method of key points. Be sure to label key points and show at least two cycles. Use the graph to determine the domain and the range of each function. $$ y=\cos (4 x) $$

Short answer

The domain is all real numbers. The range is [-1, 1].

Step-by-step solution

- Identify the Base Function

The base function for this exercise is the standard cosine function: \( y = \cos(x) \).

- Understand the Transformation

The given function \( y = \cos(4x) \) includes a transformation. Specifically, the coefficient 4 inside the cosine function argument implies a horizontal compression by a factor of 4.

- Determine the Period

The period of the base cosine function is \( 2\pi \). The period of the transformed function \( y = \cos(4x) \) can be found using the formula: \( \text{Period} = \frac{2\pi}{4} = \frac{\pi}{2} \).

- Find Key Points

For the cosine function \( y = \cos(4x) \), identify key points over one period \( \frac{\pi}{2} \): \( (0, 1), \left(\frac{\pi}{8}, 0\right), (\frac{\pi}{4}, -1), \left(\frac{3\pi}{8}, 0\right), (\frac{\pi}{2}, 1) \).

- Plot Two Cycles

Plot the key points for two periods (\( 0 \leq x \leq \pi \)) using the coordinates from step 4. Connect these points smoothly to complete the graph of \( y = \cos(4x) \).

- Determine the Domain

The domain of \( y = \cos(4x) \) is all real numbers: (-\infty, +\infty).

- Determine the Range

The range of \( y = \cos(4x) \) is the same as the range of the base cosine function: [-1, 1].

Key concepts

Cosine Transformation

In trigonometric graphs, transformations are common and essential to understanding different variations of base functions. The base function in the given problem is the cosine function, represented as \(y = \cos(x)\). When we alter the argument of this function, we introduce transformations.
In our case, the function provided is \(y = \cos(4x)\). The coefficient 4 inside the cosine function indicates a horizontal transformation. This transformation compresses the cosine wave by a factor of 4. In simpler terms:

• Without any transformation, the cosine wave completes one cycle in the interval \([0, 2\pi]\).
• With the transformation \(y = \cos(4x)\), the wave completes one cycle in a shorter interval, specifically \([0, \frac{2\pi}{4}] = [0, \frac{\pi}{2}]\).

The graph now looks more squeezed horizontally than the base cosine function.

Period of Trigonometric Function

The period of a trigonometric function is the length of one complete cycle on the graph. For the base cosine function \(y = \cos(x)\), the period is \(2\pi\), meaning the function repeats its pattern every \(2\pi\) units.
When a horizontal transformation involving a coefficient k is applied inside the function argument — like \(y = \cos(kx)\) — the period changes according to the formula:

• New period = \(\frac{2\pi}{|k|}\)

For the function \(y = \cos(4x)\), the coefficient k is 4. Therefore, the new period is calculated as follows:
\(\text{Period} = \frac{2\pi}{4} = \frac{\pi}{2}\).
Thus, the function \(y = \cos(4x)\) completes one full cycle every \(\frac{\pi}{2}\).

Domain and Range

Understanding the domain and range of a function is crucial for graphing and interpreting it properly.
The domain of a function refers to all the possible input values (x-values) that the function can accept. For the cosine function \(y = \cos(4x)\):

• The domain is all real numbers, written in interval notation as \((−\infty, +\infty)\), since cosine is defined for all x-values.

The range of a function refers to all possible output values (y-values) that the function can produce. For the cosine function \(y = \cos(4x)\):

• The range remains consistent with the base cosine function, which varies between -1 and 1.

Hence, the range is represented in interval notation as \([-1, 1]\). This means the function oscillates between -1 and 1, regardless of the transformation applied to the x-values.