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 general form of the sine function: \(y = A \sin(Bx + C)\), where \(A\) is the amplitude, \(B\) affects the period, and \(C\) affects the phase shift.
2. Find the amplitude by taking the absolute value of the coefficient in front of the sine function: \(A = |3|\).
3. Calculate the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\). Here, \(B\) is the coefficient of \(x\) inside the sine function, which is \(\pi\).
4. Determine the phase shift using the formula \(\text{Phase shift} = -\frac{C}{B}\). In this case, \(C = 2\) and \(B = \pi\).
5. To graph one period of the function, start at the phase shift on the x-axis, then plot points over one full period length, using the amplitude to mark the maximum and minimum values of the sine wave.

Key Concepts

Amplitude of a Sine Function

Amplitude is the maximum value the sine function attains from its midline, representing the height of its peaks. For y = a sin(bx + c), the amplitude is the absolute value of 'a'. In this case, the amplitude is |3| = 3, indicating the wave oscillates 3 units above and below the midline.

Period of a Sine Function

The period is the length of one complete cycle of the sine wave. It is calculated as (2π) divided by the absolute value of the coefficient 'b' in y = a sin(bx + c). Here, with b = π, the period is 2π/π = 2, meaning the function repeats every 2 units along the x-axis.

Phase Shift of a Sine Function

Phase shift refers to the horizontal translation of the sine curve and is found by solving (bx + c) = 0 for x, giving -c/b. For y = 3 sin(πx + 2), the phase shift is -2/π, indicating the graph shifts left by 2/π units. This affects where the wave starts on the x-axis.