Determine the amplitude and period of each function. Then graph one period of the function. y = (1/2) sin (π/3) x

Verified step by step guidance

1. Identify the general form of the sine function: \(y = A \sin(Bx)\), where \(A\) is the amplitude and \(B\) affects the period.
2. Determine the amplitude \(A\) by taking the absolute value of the coefficient in front of the sine function. Here, \(A = \left| \frac{1}{2} \right|\).
3. Find the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\), where \(B\) is the coefficient of \(x\) inside the sine function. In this case, \(B = \frac{\pi}{3}\).
4. Calculate the period by substituting \(B\) into the formula: \(\text{Period} = \frac{2\pi}{\frac{\pi}{3}}\) (do not simplify the fraction, just set up the expression).
5. To graph one period of the function, plot the sine curve starting at \(x=0\) and ending at \(x\) equal to the period found in the previous step, using the amplitude to mark the maximum and minimum values on the \(y\)-axis.

Key Concepts

Amplitude of a Sine Function

Amplitude is the maximum absolute value of the sine function's output, representing the height from the midline to the peak. For y = (1/2) sin(π/3 x), the amplitude is the coefficient 1/2, indicating the wave oscillates between -1/2 and 1/2.

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 coefficient of x inside the sine function. Here, the period is 2π ÷ (π/3) = 6, meaning the function repeats every 6 units along the x-axis.

Graphing One Period of a Sine Function

Graphing one period involves plotting the sine curve from 0 to the period length, marking key points such as the start, maximum, midline, minimum, and end. For y = (1/2) sin(π/3 x), plot from x = 0 to x = 6, using the amplitude and period to shape the wave accurately.