Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.
y = 3 sin(πx + 2)

Verified step by step guidance

1. Identify the standard form of the sine function: \( y = a \sin(bx + c) + d \).
2. Determine the amplitude by identifying the coefficient \( a \). Here, \( a = 3 \), so the amplitude is \( |3| = 3 \).
3. Find the period of the function using the formula \( \frac{2\pi}{b} \). In this case, \( b = \pi \), so the period is \( \frac{2\pi}{\pi} = 2 \).
4. Calculate the phase shift using \( \frac{-c}{b} \). Here, \( c = 2 \) and \( b = \pi \), so the phase shift is \( \frac{-2}{\pi} \).
5. Graph one period of the function by starting at the phase shift and using the amplitude and period to plot key points, such as the maximum, minimum, and intercepts.

Key Concepts

Amplitude

Amplitude refers to the maximum height of a wave from its central axis. In the function y = 3 sin(πx + 2), the amplitude is represented by the coefficient of the sine function, which is 3. This means the graph will oscillate between 3 and -3, indicating how far the wave reaches above and below its midline.

Period

The period of a trigonometric function is the distance along the x-axis for one complete cycle of the wave. For the function y = 3 sin(πx + 2), the period can be calculated using the formula 2π divided by the coefficient of x inside the sine function. Here, the period is 2, meaning the function will complete one full cycle over an interval of 2 units along the x-axis.

Phase Shift

Phase shift refers to the horizontal shift of the graph of a trigonometric function. In the function y = 3 sin(πx + 2), the phase shift can be determined by setting the inside of the sine function equal to zero. Solving πx + 2 = 0 gives a phase shift of -2/π, indicating that the graph is shifted to the left by this amount.