91Ó°ÊÓ

When it comes to trigonometric functions like sine and cosine, amplitude plays a key role in determining how high and low the function will go. Amplitude is essentially the height of the wave, measured from the middle of the wave to its peak. For the function given in the exercise, which is \(y = -3 \sin(2 \pi x) + 2\), the amplitude is marked by the coefficient in front of the sine function.
\(-3\) is the amplitude, but the negative sign indicates that the sine wave is reflected over the x-axis, starting its cycle from a lower point rather than the standard starting phase. In this instance:

• The wave oscillates between \(-3\) and \(3\) if considering it flipped around vertically.
• The absolute value of the amplitude, \(|-3|\), is \(3\).

The amplitude tells us the extent of this oscillation from its midline, which is crucial for drawing and understanding sine waves.

The period of a sinusoidal function is the length of one complete cycle of the wave. It tells us the distance over which the wave repeats itself. For our function \(y = -3 \sin(2 \pi x) + 2\), the period is directly influenced by the factor that multiplies the \(x\) inside the sine function.
The formula to calculate the period is \(\frac{2\pi}{B}\), where \(B\) is the coefficient of the \(x\). In this problem, \(B = 2\pi\), so the period is calculated as:

• \(\frac{2\pi}{2\pi} = 1\).

This means:

• Every complete wave of this function is compressed into an interval of \(1\) unit on the x-axis.
• Understanding the period helps in plotting the function, ensuring the waves align correctly over each segment of the x-axis.

Since this period is shorter than the standard sine wave, it means the waves are more frequent and repeat more quickly.

In sinusoidal functions, a vertical shift moves the graph up or down along the y-axis without affecting its shape. For the function \(y = -3 \sin(2 \pi x) + 2\), the \(+2\) at the end indicates a vertical shift.
This shift changes the centerline from which the wave oscillates, moving it upwards by 2 units. This adjustment affects the vertical alignment by:

• Raising the middle of the wave from \(y = 0\) to \(y = 2\).
• Adjusting the oscillation range to \([ -1, 5 ]\), since the whole graph is shifted upwards.

Vertical shifts are essential for positioning the function correctly in a graph, ensuring it aligns with real data or a specific desired shape. With this function, the shift makes the wave's motion intuitive to visualize, accurately reflecting its adjusted range.