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The sine function is one of the fundamental waves in trigonometry. Represented mathematically as \( \sin x \), it oscillates between -1 and 1. This makes the sine function periodic, meaning it repeats its values in regular intervals. Each cycle of the sine wave spans an interval from 0 to \( 2\pi \), where \( \pi \) (pi) is approximately 3.14159. In the function \( y = \sin x \), the highest value (\

Amplitude and period are crucial features of trigonometric functions. The amplitude measures the peak deviation of the wave from its central axis. For the basic sine function \( y = \sin x \), the amplitude is 1. This means the wave reaches 1 unit above and 1 unit below its central axis.
The period is the length it takes for the wave to complete one full cycle. For \( y = \sin x \), this period is \( 2\pi \). This implies the function repeats itself every \( 2\pi \). For our specific function, \( y = 1 + \sin x \), the amplitude is unchanged, but the vertical shift alters its central line.
The amplitude and period provide valuable insights into the behavior and shape of the wave, guiding us in plotting its graph accurately. By understanding these traits, we can easily predict the wave's oscillation and frequency.

• Amplitude: 1 (given by the coefficient of \( \sin x \))
• Period: \( 2 \pi \) (wave completes one full cycle every \( 2 \pi \) units)

The vertical shift moves the entire graph of a function up or down without altering its shape. In \( y = 1 + \sin x \), the +1 term induces a vertical shift.
This means the entire sine wave is moved up by 1 unit. As a result, the midline (central axis) of the wave is no longer at y = 0 but has shifted to y = 1.
Given this vertical shift, we can still utilize the sine wave's properties but adjust all values accordingly:

• When y = 0 for \( \sin x \), the graph of \( y = 1 + \sin x \)
• At its maximum, \( \sin x \) reaches 1, but in \( y = 1 + \sin x \)
• At its minimum, \( \sin x \) hits -1, but now this is shifted to 0 in the graph of \( y = 1 + \sin x \)

From these observations, we can meticulously draw the adjusted sine wave with its new vertices, retaining its characteristic smooth, wave-like curve.