Draw the graphs of y = sin⁡(x−1), where resresent the fractional part function.

To graph the function \( y = \sin(x - 1) \), we first need to understand its components. The sine function oscillates between -1 and 1, and the expression \( x - 1 \) indicates a horizontal shift to the right by 1 unit.

Key Features of the Graph

• Amplitude: The maximum value is 1 and the minimum is -1.
• Period: The period of the sine function is \( 2\pi \), meaning it repeats every \( 2\pi \) units.
• Phase Shift: The graph is shifted to the right by 1 unit due to the \( -1 \) in the argument.

Steps to Draw the Graph

1. Start by plotting the basic sine wave, which starts at (0, 0), reaches (π/2, 1), (π, 0), (3π/2, -1), and returns to (2π, 0).
2. Shift all these points to the right by 1 unit. For example, (0, 0) becomes (1, 0), (π/2, 1) becomes (π/2 + 1, 1), and so on.
3. Continue this pattern for additional cycles of the sine wave.

Graph Characteristics

The resulting graph will show a smooth wave that oscillates between -1 and 1, starting from the point (1, 0). It will continue to repeat this pattern indefinitely in both directions along the x-axis.

Visual Representation

While I can't draw the graph here, you can use graphing software or a graphing calculator to visualize it. Simply input the function \( y = \sin(x - 1) \) to see the wave pattern clearly.