Problem 2 - Algebra 2 and Trigonometry

Question

Is the graph of \(y=\cos x\) its own image under the translation \(T_{-2 \pi, 0} ?\) Justify your answer.

Step-by-Step Solution

Answer

Yes, it is its own image because the cosine function is periodic with a period of \(2\pi\).

Step 1: Understanding the translation

The translation given is \( T_{-2\pi, 0} \). This means that the graph is shifted horizontally to the left by \( 2\pi \) units, and there is no shift vertically. Essentially, we need to determine whether shifting the graph left by \( 2\pi \) units results in a graph that looks exactly like the original \( y = \cos x \).

Step 2: Properties of the cosine function

The cosine function \( y = \cos x \) is periodic with a period of \( 2\pi \). This means that the graph repeats itself every \( 2\pi \) units. In other words, for any value \( x \), \( \cos(x) = \cos(x + 2\pi) \). This periodicity will play a crucial role in determining if the translated graph matches the original graph.

Step 3: Application of periodicity

By shifting the graph of \( y = \cos x \) to the left by \( 2\pi \) units, we are moving each point \( (x, \cos x) \) to \( (x - 2\pi, \cos(x - 2\pi)) \). However, using the periodicity of cosine, we have \( \cos(x - 2\pi) = \cos x \). Therefore, every point on the graph after the translation matches a corresponding point on the original graph.

Step 4: Conclusion

Since the graph's points before and after the translation are equivalent due to the periodicity of the cosine function, the graph of \( y = \cos x \) is indeed its own image under the translation \( T_{-2\pi, 0} \).

Key Concepts

Graph Translation

Graph translation involves moving a graph from one position to another without altering its shape or orientation. In mathematical terms, this is achieved by shifting the graph horizontally or vertically in the coordinate plane. For example, considering the graph of the function \( y = \cos x \), a translation \( T_{-2\pi, 0} \) denotes a horizontal shift to the left by \( 2\pi \) units, without any vertical displacement. When you perform a translation, you change the location of each point on the graph. However, the graph's shape remains consistent with its original form, preserving the function's properties. Translations are quite straightforward but essential for analyzing how functions behave under these shifts, especially when exploring properties like periodicity.

Periodic Function

Periodic functions repeat their values in regular intervals across the domain. This characteristic is most often associated with trigonometric functions, such as the cosine and sine functions.For the cosine function \( y=\cos x \), its periodicity is expressed as having a period of \( 2\pi \). This signifies that every \( 2\pi \) units along the x-axis, the pattern of the graph repeats itself. You can imagine it like a wave that cycles identically over equal distances.A deep understanding of periodicity allows you to determine how functions like \( y=\cos(x) \) respond to translations. If you shift the graph left or right by a multiple of its period, say \( 2\pi \) or \( -2\pi \), the resultant graph appears unchanged compared to the original, because it aligns perfectly with its prior positions.

Trigonometric Graphs

Trigonometric graphs, such as those of sine, cosine, and tangent, display the relationships between angles and ratios in a periodic, wave-like form. The graph of \( y=\cos x \) is one of the fundamental trigonometric graphs and exhibits a smooth, continuous wave oscillating between 1 and -1, showcasing the nature of cosine.These graphs are characterized by their amplitude, period, and frequency. For \( y = \cos x \), the amplitude is 1, reflecting the maximum height of the wave from its midline (the x-axis). The period is \( 2\pi \), indicating the length required for one complete cycle of the wave. Frequency, inversely related to the period, is how often the wave cycle occurs in a given interval.Understanding trigonometric graphs is critical for recognizing the underlying symmetries and transformations, such as translations and reflections. These properties help in accurately predicting and drawing the behavior of more complex trigonometric functions.