Graphs of the Sine and Cosine Functions

Objectives

This section covers the fundamental properties and transformations of the sine and cosine functions. Students will learn to:

• Recall the basic shapes and key features of y = sin x and y = cos x.

• Graph transformations of sinusoids: y = A \sin(Bx - C) + D and y = A \cos(Bx - C) + D.

• Read amplitude, period, phase shift, and midline from an equation (and vice versa).

Period and Zeros/Extrema

Both sin x and cos x are periodic functions with period (or 360°). Their zeros and extrema are as follows:

• sin x: Zeros at ; maximum $1x = \frac{\pi}{2} + 2n\pi-1$ at $x = \frac{3\pi}{2} + 2n\pi$.

• cos x: Zeros at ; maximum $1x = 2n\pi-1$ at $x = \pi + 2n\pi$.

Key Values of Sine and Cosine on [0, 2π]

The following tables summarize the values of sin x and cos x at key points in one period:

General Sinusoid and Transformations

The general form for a sinusoidal function is y = A \sin(Bx - C) + D or y = A \cos(Bx - C) + D. Each parameter affects the graph in a specific way:

• Amplitude: . This is the vertical stretch; the graph oscillates $|A|$ units above and below the midline.

• Period: (in degrees: ). Frequency: .

• Phase shift (horizontal shift): Shift . Right if , left if . Tip: Always factor first, so .

• Midline: . This is a vertical shift by .

• Reflections: reflects across the midline (vertical reflection). reflects horizontally.

Example: Constructing a Cosine Function from Parameters

Given the following parameters:

• Amplitude:

• Period:

• Phase shift:

The required form is:

Graphing Sinusoidal Functions

To graph a function such as f(x) = 5 \cos(x):

• Identify , , , .

• One period of can be defined between and .

• The -coordinates that divide this distance into four equal sub-intervals are , , and .

• Calculate the corresponding -values using , where (the values of at the -coordinates).

Example: Graph of a Sinusoidal Function

The graph below shows a sinusoidal function with amplitude, period, and midline clearly visible. The function oscillates between its maximum and minimum values, demonstrating the effect of amplitude and period.

Summary Table: Sinusoidal Function Parameters

Practice and Applications

• Given a sinusoidal equation, identify amplitude, period, phase shift, and midline.

• Graph sinusoidal functions using these parameters.

• Apply transformations to model real-world periodic phenomena, such as sound waves or seasonal temperature changes.