Graph each function over a one-period interval. See Example 3.
y = (3/2) sin [2(x + π/4)]

Verified step by step guidance

1. Identify the general form of the sine function given: \(y = \frac{3}{2} \sin \left[ 2 \left( x + \frac{\pi}{4} \right) \right]\). This matches the form \(y = A \sin(B(x - C))\), where \(A\) is the amplitude, \(B\) affects the period, and \(C\) is the horizontal shift.
2. Determine the amplitude \(A\), which is the coefficient in front of the sine function. Here, \(A = \frac{3}{2}\). This means the graph will oscillate between \(\frac{3}{2}\) and \(-\frac{3}{2}\).
3. Find the period of the function using the formula \(\text{Period} = \frac{2\pi}{|B|}\). Since \(B = 2\), the period is \(\frac{2\pi}{2} = \pi\). This means the function completes one full cycle over an interval of length \(\pi\).
4. Identify the phase shift (horizontal shift) from the expression inside the sine function. The function is \(\sin \left[ 2 \left( x + \frac{\pi}{4} \right) \right]\), which can be rewritten as \(\sin \left[ 2x + \frac{\pi}{2} \right]\). To express in the form \(\sin(B(x - C))\), factor out \(B\): \(\sin \left[ 2 \left( x - \left(-\frac{\pi}{4} \right) \right) \right]\). So the phase shift is \(-\frac{\pi}{4}\), meaning the graph shifts left by \(\frac{\pi}{4}\).
5. To graph over one period, choose the interval for \(x\) starting at the phase shift and extending one period length: from \(x = -\frac{\pi}{4}\) to \(x = -\frac{\pi}{4} + \pi = \frac{3\pi}{4}\). Plot key points within this interval by evaluating the sine function at values where the sine function typically takes values 0, 1, 0, -1, and 0 (e.g., at \(x = -\frac{\pi}{4}\), \(x = 0\), \(x = \frac{\pi}{4}\), \(x = \frac{\pi}{2}\), and \(x = \frac{3\pi}{4}\)), then multiply by the amplitude \(\frac{3}{2}\).

Key Concepts

Sine Function and Its Properties

The sine function is a periodic trigonometric function with a standard period of 2π. It oscillates between -1 and 1, and its graph is a smooth wave. Understanding its amplitude, period, phase shift, and vertical shift is essential for graphing transformations.

Amplitude and Vertical Stretch

Amplitude refers to the height of the wave from the midline to its peak. In the function y = (3/2) sin(...), the amplitude is 3/2, meaning the sine wave is vertically stretched by a factor of 1.5, increasing the maximum and minimum values accordingly.

Period and Phase Shift

The period of a sine function is given by 2π divided by the coefficient of x inside the sine. Here, the coefficient is 2, so the period is π. The phase shift is the horizontal shift caused by the addition inside the argument, here -π/4, shifting the graph left by π/4 units.