Problem 2 - Algebra 2 and Trigonometry

Question

Is the graph of \(y=\sin x\) symmetric with respect to the translation \(T_{-2 \pi, 0} ?\) Justify your answer.

Step-by-Step Solution

Answer

Yes, the graph of \(y = \sin x\) is symmetric with respect to the translation \(T_{-2\pi, 0}\) due to its periodicity.

Step 1: Understanding the Translation

The translation given is \(T_{-2\pi, 0}\), which means shifting the graph \(-2\pi\) units to the left. In functional terms, this changes the input \(x\) to \(x + 2\pi\). Our task is to check whether this translation results in symmetry with respect to the original graph of \(y = \sin x\).

Step 2: Apply the Translation

The translated function \(f(x)\) can be written as \(f(x) = \sin(x + 2\pi)\). We need to find the value of this function and compare it with the original function \(y = \sin x\).

Step 3: Evaluate the Translated Function

Since \(\sin(x + 2\pi) = \sin x\), the function \(f(x)\) is equivalent to the original function. This is due to the periodicity of the sine function with period \(2\pi\), where adding any multiple of \(2\pi\) does not change the value of the function.

Step 4: Determine Symmetry

By definition, a graph is symmetric with respect to a transformation if applying the transformation does not change the graph. Here, since \(f(x) = \sin(x + 2\pi) = \sin x\), the graph is unchanged and thus symmetric with respect to the translation \(T_{-2\pi, 0}\).

Key Concepts

Graph of Sine Function

The sine function, denoted as \(y = \sin x\), is a fundamental concept in trigonometry. It is a periodic wave-like graph that represents the sine values for angles in a unit circle. This graph oscillates between a maximum of 1 and a minimum of -1, displaying a regular up-and-down pattern.

The graph of the sine function has key features that are essential for understanding its behavior. The sine curve starts at the origin, moves up to its peak of 1 at \(\frac{\pi}{2}\), returns to 0 at \(\pi\), hits its lowest point at -1 at \(\frac{3\pi}{2}\), and comes back to 0 at \(2\pi\). This consistent pattern repeats every \(2\pi\) units along the x-axis, forming the basis for its periodic nature.

Function Translation

Translation of a function involves shifting the original graph horizontally or vertically on the coordinate plane. For the sine function \(y = \sin x\), a horizontal translation involves adjusting the input of the function.

In this context, when we apply a translation of \(T_{-2\pi, 0}\), it means moving the sine graph \(-2\pi\) units to the left. Mathematically, this adjusts the function to \(f(x) = \sin(x + 2\pi)\). This type of transformation helps in visualizing how shifts affect the overall appearance and position of the graph without altering its shape. It retains the original properties, as shown by the equivalence \(\sin(x + 2\pi) = \sin x\), due to periodicity.

Periodicity of Sine Function

The sine function is known for its periodicity; this means the function repeats its values in regular intervals. The period of the standard sine function is \(2\pi\).

Periodicity is an important property because it indicates that the values of the sine function recur at intervals of \(2\pi\). Mathematically, this can be expressed as \(\sin(x) = \sin(x + 2\pi n)\), where \(n\) is any integer. This repeating nature reflects the circular motion of angles measured in radians, influencing how transformations such as translations affect the graph.

• Every \(2\pi\) units along the x-axis, the sine function completes one full cycle.
• This property makes the sine graph appear seamless even after shifts, rotations, or translations by multiples of \(2\pi\).

Understanding periodicity is crucial when analyzing symmetries and transformations of trigonometric graphs.

Mathematical Symmetry

Mathematical symmetry in functions refers to the invariance or unchanged state under specific transformations. For the sine function, symmetry can be assessed through various operations, including translations.

In the problem context, translating the sine graph by \(-2\pi\) units results in no change, demonstrating symmetry. Since \(\sin(x + 2\pi) = \sin x\), the function retains its original form after the given translation.

• Symmetry with respect to transformations means the graph looks identical before and after the applied operation.
• For sine, this includes shifts along the x-axis by multiples of \(2\pi\).

This characteristic is due to the intrinsic periodicity of the sine function, which ensures that the value at any angle plus \(2\pi\) is the same as at the original angle.