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Amplitude in trigonometric functions tells us about the height of the wave. It's the distance from the middle of the wave to its peak, which shows us how tall the wave gets from the midline.
To determine the amplitude, we look at the coefficient of the trigonometric function, usually found in front of sine or cosine, like in the function \(y = A \sin(Bx + C) + D\). Here, \(A\) represents the amplitude.

• If \(A\) is positive, the wave reaches upward from its midline.
• If \(A\) is negative, the wave starts by moving downward first.

In our case, the function is \(y = \sin \frac{1}{2}(x + \frac{\pi}{4})\), where the coefficient of the sine function—the amplitude—is \(1\). This means the wave extends 1 unit both above and below the midline at \(y=0\), making its peaks and troughs equally spaced.

The period of a trigonometric function tells us the length it takes for the wave to complete one full cycle. Essentially, it’s how far you move along the x-axis before the wave starts repeating.
For sine and cosine functions, the period is determined using the formula \(\frac{2 \pi}{|B|}\), where \(B\) is the coefficient of \(x\) inside the function (found in \(y = A \sin(Bx + C) + D\)).

• A larger \(B\) value results in a shorter period because the wave squashes more cycles into the same distance.
• A smaller \(B\) stretches the wave out, giving a longer period.

In the function \(y = \sin \frac{1}{2}(x + \frac{\pi}{4})\), \(B\) is \(\frac{1}{2}\). This makes the period \(\frac{2 \pi}{0.5} = 4 \pi\), meaning every 4Ï€ units along the x-axis, the sine wave repeats its cycle. So, the function completes one full pattern every \(4 \pi\) units.

The phase shift in a trigonometric function indicates horizontal translation—essentially, how far the function moves left or right from its standard starting position.
It’s dictated by the \(C\) value in the equation \(y = A \sin(Bx + C) + D\). A positive \(C\) value shifts the wave left, while a negative one shifts it right (since it’s inside the expression \((Bx + C)\)).
To find the phase shift, use the formula \(-\frac{C}{B}\). This helps determine how much and in what direction the shift occurs.

• A move to the left occurs if \(\frac{C}{B}\) results in a negative value.
• A move to the right occurs with a positive \(\frac{C}{B}\) value.

For \(y = \sin \frac{1}{2} (x + \frac{\pi}{4})\), \(C\) is \(\frac{\pi}{4}\), causing a shift of \(-\frac{\pi/4}{0.5} = \text{-} \frac{\pi}{2}\) or \(-\frac{\pi}{4}\) to the left. The wave starts its cycle earlier by that amount on the x-axis.