91Ó°ÊÓ

The sine function, denoted as \(\text{sin}(x)\), is a foundational trigonometric function. It depicts oscillatory motion, commonly appearing in waves and cycles. The general sine function can be written as \(\text{y} = \text{A} \text{sin}(\text{B}(x - C)) + D\), where each parameter influences the function's shape and position:

• \(\text{A}\): Amplitude, dictating the wave's height.
• \(\text{B}\): Controls the period, determining how quickly the function completes one cycle.
• \(\text{C}\): Phase shift, moving the wave left or right.
• \(\text{D}\): Vertical translation, shifting the wave up or down.

In our given exercise, the sine portion is represented by \( \text{sin}(2(x + \frac{\text{Ï€}}{4} ))\), indicating some modifications to the standard function.

The period of a trigonometric function represents the length of one complete cycle. For sine functions, the period is influenced by the coefficient \( \text{B} \) as seen in \( \text{y} = \text{A} \text{sin}(\text{B}(x - C)) + D \). Specifically, the period is calculated using the formula \(\frac{2\text{Ï€}}{\text{B}}\). In our exercise, the coefficient inside the sine function is 2: \(\text{sin}(2(x + \frac{\text{Ï€}}{4}))\). Thus, the period is calculated as: \[\frac{2\text{Ï€}}{2} = \text{Ï€}\text{.} \] This indicates that one complete cycle of this sine function occurs over an interval of \( \text{Ï€} \) units along the x-axis, causing the wave to complete more cycles within the same space compared to a standard sine function.

A phase shift moves the graph of a sine function left or right. It is determined by the horizontal shift \( \text{C} \), found within the parenthesis of the sine function: \(\text{y} = \text{A} \text{sin}(\text{B}(x - C)) + D \). In our function, \( \text{sin}(2(x + \frac{\text{Ï€}}{4})) \), there's an apparent rightward shift due to the positive term inside the parentheses. To find the phase shift, we calculate: \(- \frac{\text{Ï€}}{4} \). This phase shift tells us the graph moves left by \(\frac{\text{Ï€}}{4} \) units. Understanding phase shifts is crucial for accurately plotting the function's start and key points.

Vertical translation shifts the entire sine function up or down on the y-axis. This movement is governed by the constant \( \text{D} \) in the general function \( \text{y} = \text{A} \text{sin}(\text{B}(x - C)) + D \). In our function, the term outside of the sine is \( \frac{1}{2} \), leading to an upward shift by \( \frac{1}{2} \) units. This means every point on the sine wave moves up by \( \frac{1}{2} \). Understanding vertical translations aids in positioning the graph and ensuring each calculated point's y-value is adjusted accordingly.

Graphing a sine function involves identifying key points like intercepts, maximum, and minimum values. For \( y = \frac{1}{2} + \text{sin}(2(x + \frac{\text{Ï€}}{4})) \), we first plot the sine function \( \text{sin}(2x) \) over one period \( [0, \text{Ï€}] \). Then, we shift it left by \( \frac{\text{Ï€}}{4} \) and move all points up by \( \frac{1}{2} \):

• Start: \( y = \frac{1}{2} + \text{sin}(-\frac{\text{Ï€}}{2}) = 0 \)
• Quarter period: \( y = \frac{1}{2} + \text{sin}(0) = \frac{1}{2} \)
• Half period: \( y = \frac{1}{2} + \text{sin}(\frac{\text{Ï€}}{2}) = 1 \)
• Three-quarter period: \( y = \frac{1}{2} + \text{sin}(\text{Ï€}) = \frac{1}{2} \)
• End: \( y = \frac{1}{2} + \text{sin}(3\frac{\text{Ï€}}{2}) = 0 \)

Plotting these points helps illustrate the function and captures all transformations effectively.