Short Answer 
The parent sine function, defined as y = sin(x), has a midline of y = 0, an amplitude of 1, and a period of 2œA. To modify the sine function for specific features, the general form y = Asin(Bx + C) + D adjusts the midline, amplitude, and period, allowing for custom graphs such as y = 3sin(œAx/2) Р1, which features a specific period and amplitude.
Step 1: Understanding the Basic Sine Function
The parent sine function is defined as y = sin(x). It has specific characteristics including:
- Midline: The value where the function oscillates, which is y = 0.
- Maximum: The highest point the function reaches, which is 1.
- Minimum: The lowest point, which is -1.
- Amplitude: The height from the midline to the maximum or minimum, which equals 1.
- Period: The length of one complete cycle, which is 2œA.
Step 2: Modifying the Sine Function for Custom Features
The general form of the sine function can be expressed as y = Asin(Bx + C) + D, where each component modifies the behavior:
- Midline: Vertical shift, represented as y = D.
- Maximum: Recalculated as A + D.
- Minimum: Calculated as -A + D.
- Amplitude: The height from the midline, defined as A.
- Period: This is adjusted by 2œA/B.
- Phase Shift: The horizontal shift is defined by C.
Step 3: Constructing and Graphing the Sine Function
With specific values assigned, you can form the sine function you need to graph:
- Period: Given as 4, find B = œA/2.
- Amplitude: Set as 3.
- Midline: Adjusted to -1, hence D = -1.
- Using the known y-intercept, the function becomes y = 3sin(œAx/2) Р1.
- Maxima and minima points should be calculated as 2 and -4 respectively, while other key points like intercepts are also identified to accurately plot the function.