Problem 16 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Identify the amplitude and period of the function. Then graph the function and describe the graph of \(g\) as a transformation of the graph of its parent function. \(g(x)=\cos 4 x\)

Step-by-Step Solution

Answer

The amplitude of the function \(g(x) = \cos 4x\) is 1 and the period is \(\frac{\pi}{2}\). The graph of \(g(x)\) is a horizontally compressed version of the graph of \(\cos x\) by a factor of 1/4.

Step 1: Identify the amplitude

The amplitude of a function is the absolute value of the coefficient of the function. In this case, since there is no coefficient in front of the \(\cos\) function, the amplitude is 1.

Step 2: Find the period

The period of a function is determined by the coefficient of the variable inside the function. It can be calculated using the formula \( \frac{2\pi}{|B|}\), where B is the coefficient of the variable x. For this function, the period will be \( \frac{2\pi}{4} = \frac{\pi}{2}\).

Step 3: Graphing the function

The function \(g(x) = \cos 4x\) will make four full cycles in the span of \(2\pi\) because it is horizontally compressed by a factor of 1/4. To graph, start by marking the period, amplitude and midline. Then mark one cycle, from peak to peak, within the length of the period. Being a cosine function, it starts with a peak.

Step 4: Describe the transformation

The function \(g(x) = \cos 4x\) is a transformation of the parent function \(\cos x\). The graph of \(g(x)\) is a horizontally compressed version of the graph of \(\cos x\) by a factor of 1/4. There is no vertical transformation since the amplitude remains the same.

Key Concepts

Amplitude

The amplitude of a trigonometric function is a measure of how much the function oscillates above and below its midline. Imagine it as the height of the wave produced by the function.

For cosine and sine functions, which are periodic, the amplitude determines the distance from the midline to the highest or lowest point on the graph.

• If a function is written as \[ y = a \, \cos(bx) \] or \[ y = a \, \sin(bx) \]
• The amplitude is \( |a| \).

In the given function \( g(x) = \cos 4x \), the coefficient \( a \) of the cosine function is implicit as 1. Therefore, the amplitude is \( |1| = 1 \). This means the function's graph will oscillate 1 unit above and 1 unit below the midline of the graph.

Period of a Function

The period of a trigonometric function is the horizontal length over which the function completes one full cycle of its pattern. It's like the interval after which the function starts repeating itself.

To find the period of functions like cosine and sine, we use the formula:

• \[ Period = \frac{2\pi}{|b|} \]

Here, \( b \) is the coefficient of \( x \) inside the cosine or sine function.

For the function \( g(x) = \cos 4x \), the coefficient \( b \) is 4. This gives us a period of: \[\text{Period} = \frac{2\pi}{4} = \frac{\pi}{2}\]
This means the cosine wave completes one full cycle every \( \frac{\pi}{2} \) units. Thus, within the interval \([0, 2\pi]\), the function \( g(x) \) will complete four cycles.

Graph Transformations

Graph transformations involve changing the position, orientation, or size of the graph of a function. These transformations are vital when modifying the shape and position of graphs to understand how equations behave visually.

When dealing with the function \( g(x) = \cos 4x \), we are interested in transformations from the parent function \( \cos x \). In this case:

• The graph is horizontally compressed.
• Normally, \( \cos x \) spans a complete wave within \( 2\pi \), but \( \cos 4x \) now does so in \( \frac{\pi}{2} \).

This compression is due to the factor of 4 multiplying the \( x \) inside the cosine function.

Another transformation aspect to observe includes the starting point. While the amplitude remains 1, showing no vertical stretch or shrink, the wave still begins at a maximum, consistent with the parent cosine graph. No vertical or horizontal shifts occur, maintaining the symmetry of the cosine's peaks and troughs. Thus, \( g(x) = \cos 4x \) is essentially a compressed version of the original \( \cos x \) graph.