Problem 15 - Precalculus : Building Concepts and Connections

Question

Use your knowledge of horizontal translations to graph at least two cycles of the given functions. $$g(x)=\cos \left(\frac{3 \pi}{4}+x\right)$$

Step-by-Step Solution

Answer

The function \(g(x) = \cos \left(\frac{3 \pi}{4}+x\right)\) is a cosine function shifted \(-\frac{3 \pi}{4}\) units to the left. Plotting two cycles of this function starts from \(-\frac{3 \pi}{4}\) and repeats the standard shape of the cosine function until \(-\frac{3 \pi}{4} + 2*2\pi\).

Step 1: Identify the phase shift

First, we look inside the brackets of the cosine function \(\cos \left(\frac{3 \pi}{4}+x\right)\). The function is translated horizontally by \(-\frac{3 \pi}{4}\) units. So the phase shift is \(-\frac{3 \pi}{4}\).

Step 2: Plot the standard cosine graph

Draw the cosine function \(y = \cos x\). At \(x = 0\), \(cos x = 1\). The function then oscillates between -1 and 1 with a period of \(2\pi\). This means that every \(2\pi\) units, the graph repeats itself.

Step 3: Apply the phase shift

Now we apply the phase shift identified in Step 1. Every point on the graph will move \(-\frac{3 \pi}{4}\) units to the left. This gives us the graph of the function \(g(x) = \cos \left(\frac{3 \pi}{4}+x\right)\).

Step 4: Plot two cycles

To complete the exercise, we should plot at least two cycles of the function. Starting from the point \(-\frac{3 \pi}{4}\), one cycle finishes at \(-\frac{3 \pi}{4} + 2\pi\) and the second cycle at \(-\frac{3 \pi}{4} + 2*2\pi\). Plot these points and draw the graph repeat the shape of the standard cosine graph between these points.

Key Concepts

Phase Shift in Trigonometric Functions

In trigonometry, the phase shift refers to the horizontal movement of a graph along the x-axis. If you're working with a trigonometric function like the cosine function, a phase shift means altering where each cycle of the function begins on the graph.
It is important because it affects the starting point of the wave without changing its shape. For the given equation, \(g(x) = \cos \left(\frac{3 \pi}{4} + x\right)\), the expression inside the cosine, \(\frac{3 \pi}{4}+x\), indicates a phase shift.

• To determine the phase shift, look at the added or subtracted value with \(x\) inside the function.
• Here, \(\frac{3 \pi}{4}\) is added to \(x\), meaning every point on the standard cosine graph moves \(-\frac{3 \pi}{4}\) units horizontally to the left.
• The negative sign denotes a shift to the left, while a positive sign would mean a shift to the right.

Understanding phase shifts helps in predicting and accurately plotting sinusoidal graphs, keeping them aligned in the real-world or theoretical settings.

Understanding the Cosine Function

The cosine function, represented as \(y = \cos x\), is one of the fundamental trigonometric functions. It's periodic, meaning it repeats its pattern over regular intervals.
This interval, known as the period, is \(2\pi\) for the cosine function, after which the graph starts repeating itself exactly.
The cosine wave looks like a series of peaks and troughs, and its amplitude is the maximum value it reaches, which is 1, while its minimum value is -1.

• Starts at 1 when \(x = 0\) and follows a wave-like structure.
• The key points in one period are where the graph starts at its maximum, goes through zero, reaches its minimum, and returns to zero.
• It's important to note how the properties of amplitude and period are separate from the horizontal translation or phase shift.

By understanding these basic features of the cosine function, one can easily transform and transpose the wave for various practical purposes, keeping the amplitude and period intact while adjusting the phase as needed.

Graphing Trigonometric Functions with Horizontal Translations

Graphing trigonometric functions, especially with horizontal translations, involves both the basic curve features and the application of transformations like phase shifts.
To successfully graph a function such as \(g(x) = \cos \left(\frac{3 \pi}{4} + x\right)\), we employ steps to adjust for these changes.

• Firstly, identify the original graph of \(y = \cos x\), which repeats every \(2\pi\) units.
• Identify the phase shift from the function's equation, \(-\frac{3 \pi}{4}\) in this case, moving the graph to the left by this much.
• Embed this shift visually by shifting each point in the original cosine graph to the left by \(-\frac{3 \pi}{4}\) units.
• Graph at least two cycles to fully demonstrate the shifted function's behavior over a complete interval, ensuring continuity and repeatability of the waveform.

Always check axis labels and scales to ensure consistency with the transformations applied. Understanding this process helps in analyzing and synthesizing trigonometric waveforms in practical scenarios like physics and engineering.