Problem 50 - Precalculus: A Concise Course

Question

Sketch the graph of the function. (Include two full periods.) $$ y=\sin (x-2 \pi) $$

Step-by-Step Solution

Answer

After applying the shift, the function \(y=\sin(x-2\pi)\) has key points at \((2\pi, 0), (\frac{5\pi}{2}, 1), (3\pi, 0), (\frac{7\pi}{2}, -1)\), and \((4\pi, 0)\). Draw a smooth curve connecting these points to complete the graph.

Step 1: Identify Key Points

Start with the basic sine function, \(y=\sin(x)\), and find key points. The key points for a basic sine function in one period from 0 to \(2\pi\) are: (0, 0), \((\frac{\pi}{2}, 1)\), \((\pi, 0)\), \((\frac{3\pi}{2}, -1)\), and \((2\pi, 0)\).

Step 2: Apply Shift

With a right shift of \(2\pi\), the key points just found will also shift \(2\pi\) units to the right. Thus, adjusting the points gives: \((2\pi, 0), (\frac{5\pi}{2}, 1), (3\pi, 0), (\frac{7\pi}{2}, -1)\), and \((4\pi, 0)\).

Step 3: Draw the Graph

Plot these shifted key points on an x-y graph. After plotting, connect the points with a smooth curve and repeat to complete two full periods. Extend the pattern.

Key Concepts

Sine Function

The sine function, noted as \(y = \sin(x)\), is one of the fundamental trigonometric functions. It reports the y-coordinates of a point as it varies along the unit circle. In simpler terms, it tells you how high or low a point is as it travels counterclockwise from the circle’s starting point. The sine function waves up and down, creating a smooth, continuous curve.
Sine functions are central to a broad range of applications like in sound waves, light waves, and alternating current in electricity.

• Its maximum value is 1.
• The minimum value is -1.
• The sine wave starts at 0 and returns to 0 at the end of each cycle, at times known as "zero crossings".

You can visualize this as a hill and a valley that repeats, resembling the smooth undulations we often see in nature.

Periodic Functions

A function is periodic if its values repeat at regular intervals, known as the period. The sine function is a prime example of such a behavior. Its period is \(2\pi\), meaning it completes a full cycle — up and down, back to the start — every \(2\pi\) units horizontally along the x-axis.
Consider what this means:

• Every \(x\) value that increases by \(2\pi\) will give you the same \(y\) value.
• This repetition is what gives the sine wave its characteristic wave pattern.

Periodic functions are important for understanding cycles in science and nature, where processes often repeat in a predictable manner over time.

Phase Shift

Phase shift in trigonometric functions refers to the horizontal movement of a graph along the x-axis. It describes how far the graph is "shifted" from the usual starting point.
The expression \(y=\sin(x-2\pi)\) in the original exercise has a phase shift of \(2\pi\) to the right. This means:

• Each point on the sine wave moves \(2\pi\) units to the right.
• The wave begins its upward motion not at 0 on the x-axis, but instead effectively "starts" at \(2\pi\).

Phase shifts don't alter the shape of the graph. They simply reposition it along the x-axis, maintaining its features but adjusting its starting point.

Key Points

Key points on a sine graph help you understand where the function is at its extremities and zero crossings. In one period of the sine function, which spans from 0 to \(2\pi\), key points include:

• Start at (0, 0) where the wave crosses the axis.
• Reach its peak at \((\frac{\pi}{2}, 1)\).
• Fall back to 0 at \((\pi, 0)\).
• Drop to its lowest point at \((\frac{3\pi}{2}, -1)\).
• Return to 0 at the end of the period, \((2\pi, 0)\).

In the given problem, these points shift because of the phase shift:

• New start at \((2\pi, 0)\).
• New peak at \((\frac{5\pi}{2}, 1)\).
• Return to neutral at \((3\pi, 0)\).
• New low at \((\frac{7\pi}{2}, -1)\).
• And again at neutral at \((4\pi, 0)\).

These key points help you sketch the graph accurately, no matter the shifts.