Problem 2 - Algebra 2 and Trigonometry

Question

Is the graph of \(y=\cos 2\left(x+\frac{\pi}{4}\right)\) the same as the graph of \(y=\cos \left(2 x+\frac{\pi}{4}\right) ?\) Justify your answer.

Step-by-Step Solution

Answer

No, the graphs are not the same due to different shifts and compressions.

Step 1: Understand the Parent Function

The parent function we are working with is the cosine function, which is periodic and oscillates between -1 and 1.

Step 2: Identify Transformations for the First Equation

For the equation \(y = \cos 2\left(x+\frac{\pi}{4}\right)\), rewrite it as \(y = \cos \left(2x + \frac{\pi}{2}\right)\). The transformation indicates a horizontal shift by \(-\frac{\pi}{4}\) (shift to the left) and a horizontal dilation by \(\frac{1}{2}\) (compression).

Step 3: Identify Transformations for the Second Equation

For the equation \(y = \cos \left(2x + \frac{\pi}{4}\right)\), interpret it directly as a horizontal shift by \(-\frac{\pi}{8}\) (shift to the left) without affecting the period because the dilation factor is 1.

Step 4: Compare Transformations

Compare the transformations from both equations: the first is shifted by \(-\frac{\pi}{4}\) and compressed by 2, while the second is only shifted by \(-\frac{\pi}{8}\). Thus, while both involve shifts and dilations, the amounts differ, leading to different graphs.

Step 5: Conclude on Graph Equality

Since the transformations applied to each equation result in different phase shifts and compressions, the graphs of \(y = \cos 2\left(x+\frac{\pi}{4}\right)\) and \(y = \cos \left(2x+\frac{\pi}{4}\right)\) are not the same.

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It is commonly denoted as \( y = \cos(x) \). This function produces a wave-like graph that oscillates between -1 and 1. The regularity of its pattern is due to its periodicity, which repeats every \(2\pi\).

• Periodicity: The cosine function repeats its values in a regular cycle of \(2\pi\).
• Amplitude: The height of the peaks (1) and the depths of the troughs (-1) determine its amplitude.
• Key points: The function takes the value of 1 at \(x = 0\) and behaves symmetrically around this point.

Understanding these properties is crucial for effectively identifying transformations such as shifts and compressions.

Horizontal Shift

A horizontal shift occurs when the graph of a function is moved left or right. In the context of the cosine function, it involves modifying the angle inside the function. For example, if you see \( y = \cos(x + c) \), it indicates a shift:

• If \(c\) is positive, the graph shifts to the left.
• If \(c\) is negative, the graph shifts to the right.

Thus, for the equation \(y = \cos 2(x + \frac{\pi}{4})\), it indicates a shift leftward by \(\frac{\pi}{4}\). Horizontal shifts are crucial because they affect the starting location of the cosine wave along the x-axis, altering the cycle of peaks and troughs.

Phase Shift

When we talk about phase shift in trigonometric transformations, we mean the horizontal displacement of the wave. It is closely related to the horizontal shift but often involves additional transformations like dilation.

• In \(y = \cos 2(x + \frac{\pi}{4})\), after expanding, the phase shift is seen through the transformation to \(y = \cos(2x + \frac{\pi}{2})\).
• This results in a shift left by \(\frac{\pi}{4}\) and a halving of the period, compacting the wave.

Analyzing both given equations, different phase shifts are achieved due to their respective alterations to those transformations. Phase shifts are fundamental as they dictate how the function compares to its basic form.

Graph Comparison

Comparing graphs of trigonometric functions involves looking at transformations such as shifts and dilations. For the given equations, we need to compare:

• \(y = \cos 2(x + \frac{\pi}{4})\) involves a horizontal shift of \(-\frac{\pi}{4}\) and a compression by a factor of 2.
• \(y = \cos (2x + \frac{\pi}{4})\) applies a smaller shift of \(-\frac{\pi}{8}\) without compression.

Consider these transformations graphically:- The first function shifts and compresses more, drastically altering its pattern on the axis.- The second function experiences less dramatic transformation, maintaining more of its initial rhythm.From this comparison, it becomes clear the graphs are different and each transformation uniquely alters the shape and position of the cosine wave in relation to its original form.