5.3 Sinusoidal Models: Period, Amplitude, and Phase

Why sinusoidal models are different

A sinusoidal model describes a quantity that rises and falls in a repeating cycle. Daylight hours, seasonal temperature, tides, Ferris wheel height, and alternating current all have this shape when the context is appropriate. The Mathematics 30-2 formula sheet includes the form y = a sin(bx + c) + d and the period relationship, and the bulletin notes that students often struggle to connect sinusoidal characteristics to the context of a problem. That is the key skill: not just naming amplitude, but explaining what the amplitude means in the situation.

A sine or cosine graph repeats, so many equations can model the same graph. Math 30-2 usually rewards correct features and reasonable equations rather than one memorized template. Your first job is to locate maximum, minimum, midline, and period. After that, choose sine or cosine based on a convenient starting point.

Core features

Amplitude measures vertical distance from the midline to a maximum or minimum. If the maximum is M and the minimum is m, amplitude = (M - m) / 2 and midline = (M + m) / 2. The midline is the vertical center, so in y = a sin(bx + c) + d, the midline is y = d. The amplitude is |a|, not a if a is negative. A negative a reflects the graph across the midline; it does not make amplitude negative.

Period is the horizontal length of one full cycle. If two consecutive peaks occur at x = 3 and x = 15, the period is 12. Frequency is cycles per unit, so frequency = 1 / period. If a problem says an event repeats every 24 hours, the period is 24 hours and the frequency is 1/24 cycle per hour.

Period and phase

In radian form, the period of y = a sin(bx + c) + d is 2pi / |b|. In degree form, it is 360 / |b|. Many context models avoid angle language by writing b = 2pi / period. For example, if a cycle takes 24 hours, a convenient coefficient is b = 2pi/24 = pi/12 when using radians.

Phase shift is the horizontal movement of the graph. In y = a sin(b(x - h)) + d, the shift is h units to the right. In y = a sin(bx + c) + d, the shift is -c/b. Students often lose the sign because bx + c hides the transformation. Rewriting inside as b(x - h) is safer.

Worked example: daily temperature

A town's temperature is modeled over one day. The minimum is 12 C at 6:00 a.m., and the maximum is 24 C at 6:00 p.m. The period is 24 hours. The amplitude is (24 - 12) / 2 = 6 C, and the midline is (24 + 12) / 2 = 18 C. A cosine model using t as hours after midnight can start at the maximum: T(t) = 6 cos((pi/12)(t - 18)) + 18. At t = 18, the cosine input is 0, so the model gives 24 C. At t = 6, the input is -pi, so the model gives 12 C.

A sine model could also work if it uses a different phase shift. That is why written-response explanations should focus on the features and the context. The model predicts temperature, not just y-values, so the conclusion should use degrees Celsius and time.

Reading graphs and tables

From a graph, do not estimate amplitude from the x-axis unless the midline is y = 0. Use the maximum and minimum. From a table, look for a repeated pattern and equal spacing in x. If values are 18, 24, 18, 12, 18 over equal 6-hour steps, the full cycle is 24 hours. Those values show the midline crossings, maximum, midline crossing, minimum, and return to midline.

Diploma traps

The most common trap is confusing amplitude with maximum. If a daylight model has maximum 17 hours and minimum 7 hours, the amplitude is 5 hours, not 17 hours. Another trap is treating period and frequency as the same. A shorter period means more cycles per unit, so frequency increases when period decreases.

Phase shift questions also invite over-calculation. If the question asks for the first maximum shown on a graph, read the x-coordinate of the peak. Do not build a full equation unless the question demands it. When a context is given, state what the peak, trough, and midline mean. The official bulletin's comment about contextual sinusoidal difficulty is a reminder that interpretation is part of the mathematics.

Building an equation from features

When an equation is required, build it in layers. First find a and d from amplitude and midline. Next find b from the period. Last choose the shift from a convenient point, usually a maximum for cosine or a midline crossing for sine. If a graph has maximum 14, minimum 2, period 20, and a maximum at x = 5, one valid model is y = 6 cos((2pi/20)(x - 5)) + 8. The amplitude is 6, the midline is y = 8, and the coefficient 2pi/20 creates a 20-unit cycle.

Units and angle mode

Context models often use time, distance, or months as the input, not degrees on a unit circle. If your equation uses pi, the calculator must be in radian mode for numerical evaluation. If the equation uses 360 in the period relationship, degree mode is appropriate. Feature questions such as amplitude and midline usually do not depend on angle mode, but evaluating the model at a specific input does.