Question

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

Accepted Answer

Identify the general form of the sine function given: $$y = \frac{3}{2} \sin \left[ 2 \left( x + \frac{\pi}{4} ight) ight]. $$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. 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 $$-\frac{3}{2}. $$Find the period of the function using the formula $$	ext{Period} = \frac{2\pi}{|B|}. $$Since $$B = 2$$, the period is $$\frac{2\pi}{2} = \pi. $$This means the function completes one full cycle over an interval of length $$\pi. $$Identify the phase shift (horizontal shift) from the expression inside the sine function. The function is $$\sin \left[ 2 \left( x + \frac{\pi}{4} ight) ight]$$, which can be rewritten as $$\sin \left[ 2x + \frac{\pi}{2} ight]. To $$express in the form $$\sin(B(x - C))$$, factor out $$B$$: $$\sin \left[ 2 \left( x - \left(-\frac{\pi}{4} ight) ight) ight]. So $$the phase shift is $$-\frac{\pi}{4}$$, meaning the graph shifts left by $$\frac{\pi}{4}. To $$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 $$\frac{3}{2}.$$

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

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Concetti chiave

Sine Function and Its Properties

Amplitude and Vertical Stretch

Period and Phase Shift