91Ó°ÊÓ

The amplitude of a sine function reflects the maximum distance the function's values reach from its middle or equilibrium position. In terms of a formula, for a sine function represented as \(y = A \times \text{sin}(Bx - C)\), the amplitude is determined by the absolute value of \(A\). For example, in the function:

$$y = -4 \times \text{sin}\bigg( \frac{2 \text{Ï€} x}{3} - \frac{\text{Ï€}}{3} \bigg)$$

the coefficient \(A\) is \(-4\). Thus, the amplitude is \(\text{Abs}(-4)=4\). This means the graph of the sine function will oscillate 4 units above and below the horizontal axis.

The period of a sine function defines how long it takes for the function to complete one full cycle. Mathematically, for the function \(y = A \times \text{sin}(Bx - C)\), the period is calculated using the formula:

$$\text{Period} = \frac{2 \text{Ï€}}{|B|}$$

Let's take the function:

$$y = -4 \times \text{sin}\bigg( \frac{2 \text{Ï€} x}{3} - \frac{\text{Ï€}}{3} \bigg)$$

Here, \(B\) is \(\frac{2 \text{Ï€}}{3}\). Using the period formula, we have:

$$\text{Period} = \frac{2 \text{Ï€}}{\frac{2 \text{Ï€}}{3}} = 3$$

This implies that the function completes one full cycle every 3 units along the x-axis.

The phase shift of a sine function indicates how much the function is horizontally shifted from its usual position. For a function expressed as \(y = A \times \text{sin}(Bx - C)\), the phase shift is calculated by:

$$\text{Phase Shift} = \frac{C}{B}$$

Consider the function:

$$y = -4 \times \text{sin}\bigg( \frac{2 \text{Ï€} x}{3} - \frac{\text{Ï€}}{3} \bigg)$$

Here, comparing to the standard form, \(C\) is \(\frac{\text{Ï€}}{3}\) and \(B\) is \(\frac{2 \text{Ï€}}{3}\). Substituting in the formula we get:

$$\text{Phase Shift} = \frac{\frac{\text{Ï€}}{3}}{\frac{2 \text{Ï€}}{3}} = \frac{1}{2}$$

So, the phase shift is \(\frac{1}{2}\) units to the right.