Problem 15 - 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. \(g(x)=\cos 3 x\)

Step-by-Step Solution

Answer

The amplitude of the function \( g(x) = \cos 3x \) is 1 and its period is \(2\pi/3\). The graph of \( g(x) \) is the same as its parent function \( f(x) = \cos x \), but is compressed horizontally by a factor of 3.

Step 1: Identify the Amplitude and Period

The amplitude of a cosine function \( g(x) = a \cos(bx) \) is given by the absolute value of \( a \), and the period is given by \(2\pi/b\). For the given function \(g(x) = \cos 3x\), since the coefficient is 1, the amplitude is \( |1|=1 \). The period is then \(2\pi/3\).

Step 2: Graph the Function

To graph the function, one method is to plot key points and then connect them with a smooth curve. The key points should be chosen from one complete period of the function, and should include the maximum, minimum and zero points. For \( g(x) = \cos 3x \), the maximum occurs at \( x = 0 \), the minimum at the middle of the period, and the zeros at the quarter and three-quarter points of the period. Thus the key points are at \( x = 0, \pi/3, 2\pi/3, \) and \( \pi \). Connect these points with a smooth curve.

Step 3: Describe the transformation

Comparing the given function \( g(x) = \cos 3x \) with its parent function \( f(x) = \cos x \), one can see that the graph of \( g(x) \) is the same as \( f(x) \), except it's compressed horizontally by a factor of 3. This means the graph completes three cycles within \( 2\pi \) instead of one cycle in the parent function.

Key Concepts

Amplitude of Cosine Function

The amplitude of a cosine function is a measure of its 'height,' representing the maximum distance of the function's peak or trough from the function's central axis, typically the x-axis. In other words, it indicates how 'tall' the waves of the cosine curve are.

For a standard cosine function given by the equation \( g(x) = a\cos(bx) \), the amplitude is the absolute value of the coefficient \( a \). If there's no coefficient written, as in \( \cos(x) \), the amplitude is implicitly 1. In the case of the function \( g(x)=\cos 3x \), we identify the coefficient \( a \) as 1, leading to an amplitude of \( |1|=1 \).

This means if you were to graph the function, the peak of the wave would reach 1 unit above the central axis, and the trough would extend 1 unit below it. This concept is fundamental in understanding the shape and 'boldness' of the curve of trigonometric functions.

Period of Cosine Function

The period of a cosine function refers to the length of one complete cycle of the curve; after this length, the function begins to repeat itself. Understanding the period is crucial for graphing the function and examining its behavior over a given interval.

To find the period of the function \( g(x) = a\cos(bx) \), we use the formula \( 2\pi / |b| \). In our exercise, for the function \( g(x)=\cos 3x \), the value of \( b \) is 3. This gives us a period of \( 2\pi / 3 \), indicating that the graph completes its repeating pattern every \( 2\pi / 3 \) units along the x-axis. Compared to the parent cosine function \( \cos(x) \) which has a period of \( 2\pi \), our function's period is shorter, resulting in a more frequent oscillation.

Trigonometric Function Transformations

Trigonometric function transformations involve altering the basic shape of the parent function through various methods, including stretching, compressing, shifting, and reflecting.

In our exercise, comparing the function \( g(x) = \cos 3x \) with the parent function \( f(x) = \cos x \), we notice a horizontal compression. This compression is due to the coefficient of \( x \) in the argument of the cosine function. The factor of compression is the reciprocal of the coefficient \( b \) from the function's form \( g(x) = a\cos(bx) \). So, with \( b = 3 \), the graph of \( g(x) \) is horizontally compressed by a factor of \( 1/3 \), meaning it completes three cycles within the span of \( 2\pi \) where the parent function \( f(x) \) would complete just one.

Recognizing these transformations allows us to accurately sketch trigonometric functions by applying these modifications to their well-known parent functions.