In Exercises 1–4, graph one period of each function.y = 3 sin 2x

Verified step by step guidance

1. Identify the amplitude of the function. The amplitude is the coefficient of the sine function, which is 3 in this case.
2. Determine the period of the function. The period of a sine function is given by \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \). Here, \( b = 2 \), so the period is \( \frac{2\pi}{2} = \pi \).
3. Identify the phase shift. Since there is no horizontal shift in the function \( y = 3 \sin 2x \), the phase shift is 0.
4. Determine the vertical shift. There is no vertical shift in this function, so the midline is \( y = 0 \).
5. Plot the key points for one period of the sine function. Start at \( x = 0 \), then plot points at \( x = \frac{\pi}{4} \), \( x = \frac{\pi}{2} \), \( x = \frac{3\pi}{4} \), and \( x = \pi \), using the amplitude and period to determine the \( y \)-values.

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle. It is periodic, with a standard period of 2π, meaning it repeats its values every 2π units. The graph of the sine function oscillates between -1 and 1, creating a smooth wave-like pattern.

Amplitude

Amplitude refers to the height of the wave from the centerline to its peak (or trough) in a periodic function. In the function y = 3 sin 2x, the amplitude is 3, indicating that the graph will reach a maximum value of 3 and a minimum value of -3. This affects the vertical stretch of the sine wave, making it taller than the standard sine function.

Period

The period of a trigonometric function is the length of one complete cycle of the wave. For the sine function, the period can be adjusted by a coefficient in front of the variable x. In the function y = 3 sin 2x, the coefficient 2 indicates that the period is π (calculated as 2π divided by the coefficient), meaning the wave completes one full cycle over the interval from 0 to π.