Problem 25 - Precalculus : Building Concepts and Connections

Question

Use your knowledge of horizontal stretches and compressions to graph at least two cycles of the given functions. $$g(x)=\cos (3 x)$$

Step-by-Step Solution

Answer

The graph of \(g(x)=\cos(3x)\) will show two complete cycles of the standard cosine function, compressed in the \(x\)-direction by a factor of 1/3, between \(-4\pi/3\) and \(4\pi/3\).

Step 1: Identify the Standard Form of the Function

The standard form of the function is \(y=A\cos(Bx+C)+D\). Here, the function is \(g(x)=\cos(3x)\). Essentially, in this function, the value of B is 3 where B affects the period of the function.

Step 2: Compute the Period Change Due to the Coefficient

The period of a standard cosine function \(y=\cos(x)\) is \(2\pi\). When the function is \(y=\cos(Bx)\), the period changes to \(2\pi/B\). So, for the function \(g(x) =\cos(3x)\), the period will be \(2\pi/3\).

Step 3: Sketch the Graph

Plot the function \(g(x)=\cos(3x)\) on graph paper. Mark the x-axis from \(-2\pi\) to \(2\pi\). For every \(2\pi/3\), there will be one complete graph of cosine (i.e., one cycle). Thus, two cycles will be completed from \(-4\pi/3\) to \(4\pi/3\). Use values from \(-4\pi/3\) to \(4\pi/3\) to sketch the cosine curve which will appear 'compressed' due to the coefficient of 3.

Key Concepts

Understanding Horizontal Stretches

When working with trigonometric functions like the cosine function, horizontal stretches and compressions are important concepts that change the graph's appearance. A horizontal stretch occurs when you multiply the input, specifically the variable inside the cosine function, by a value less than one, elongating the graph horizontally. Conversely, a compression occurs when the multiplier is greater than one, shortening the graph horizontally.
\(g(x) = \cos(3x)\) is an example of a horizontal compression. The coefficient '3' compresses the graph, meaning that the cycles are packed closer together compared to the standard cosine wave. The period of the standard cosine wave is \(2\pi\), but this coefficient modifies the period according to the formula \(\frac{2\pi}{B}\), where B is the multiplier. This drastically changes how quickly the cycles of the graph repeat on the x-axis.
Horizontal stretches and compressions are used in various applications such as signal processing, sound wave analysis, and even in financial models where periodic data is observed.

Exploring the Cosine Function

The cosine function, denoted as \(\cos(x)\), is a fundamental trigonometric function that describes a repeating wave-like pattern. Traditionally, this function starts at \(1\), decreases to \(-1\), and then returns to \(1\) over one cycle. The pattern then repeats indefinitely, creating a smooth, periodic wave.
You can change the cosine function's appearance and characteristics using transformations: altering amplitude, changing the period, or shifting the whole wave horizontally or vertically. The amplitude is controlled by multiplying the entire cosine function by a coefficient, while the period is influenced by multiplying the x-value inside the cosine expression by a factor (as seen in \(\cos(3x)\)).
In real-world contexts, the cosine function is employed in physics for wave models, in engineering for alternating current circuits, and even in music theory to describe harmonic oscillations.

Graphing Trigonometric Functions

Graphing trigonometric functions involves understanding their properties and how modifications affect their shape and period. For the function \(g(x)=\cos(3x)\), graphing involves recognizing that it represents a standard cosine wave that has been horizontally compressed.
When beginning to graph a cosine function, start by identifying its key features: amplitude, period, phase shift, and vertical shift. For \(g(x)=\cos(3x)\), only the period is changed. Since the period is now \(\frac{2\pi}{3}\), two complete cycles of the graph occur between \(-4\pi/3\) and \(4\pi/3\).

• Compute specific x-values where the function reaches its maximum, minimum, or crosses the x-axis (zeros).
• Mark these key points on a coordinate plane.
• Draw a smooth curve through these points, observing the wave shape.

Graphing helps visualize how transformations, like horizontal compressions in \(\cos(3x)\), alter the function's periodic nature and provides insight into the patterns and behaviors of waves in various practical applications.