9.3 Graphs of Sine and Cosine Functions: Periodic Behavior, Transformations, and Applications

Ch 9.3 Graphs of the Sine and Cosine Functions

Periodic Functions

Many phenomena in mathematics and nature exhibit repeating patterns. Sine and cosine functions are classic examples of periodic functions, which repeat their values at regular intervals.

• Definition: A function f is periodic if $f(x) = f(x + np)$ for every real number x, every integer n, and some positive real number p.

• Period: The smallest positive value of p for which the function repeats is called the period.

• Example: The sine and cosine functions both have period $2\pi$.

Sine Function: Properties and Graph

The sine function is defined for all real numbers and exhibits a smooth, wave-like graph.

• Domain: $(-\infty, \infty)$

• Range: $[-1, 1]$

• X-intercepts: At $x = n\pi$, where $n$ is an integer

• Period: $2\pi$

• Symmetry: Odd function, symmetric about the origin

Cosine Function: Properties and Graph

The cosine function is similar to the sine function but starts at its maximum value.

• Domain: $(-\infty, \infty)$

• Range: $[-1, 1]$

• X-intercepts: At $x = (2n+1)\frac{\pi}{2}$, where $n$ is an integer

• Period: $2\pi$

• Symmetry: Even function, symmetric about the y-axis

Transformations of Sine and Cosine Functions

Vertical Stretch and Amplitude

Multiplying the sine or cosine function by a constant changes the height of the graph, known as the amplitude.

• Amplitude: For $y = a \sin x$ or $y = a \cos x$, the amplitude is $|a|$.

• Range: $[-|a|, |a|]$

• Example: $y = 2 \sin x$ has amplitude 2 and range $[-2, 2]$.

Horizontal Compression/Expansion and Period

Changing the coefficient of x in $y = \sin bx$ or $y = \cos bx$ alters the period of the function.

• Period Formula: $\text{Period} = \frac{2\pi}{b}$

• Example: $y = \sin 2x$ has period $\pi$ (half the standard period).

Graphing Sine and Cosine with Different Periods

To graph $y = \sin bx$ or $y = \cos bx$, follow these steps:

1. Find the period: $\frac{2\pi}{b}$

2. Divide the interval into four equal parts

3. Evaluate the function at five key points

4. Plot and connect the points with a smooth curve

Graphing y = a sin bx: Negative Amplitude

When a is negative, the graph is reflected across the x-axis.

• Example: $y = -2 \sin 3x$ is a reflection of $y = 2 \sin 3x$.

Horizontal Translations (Phase Shift)

Shifting the graph horizontally is called a phase shift. The expression $x - d$ in $y = \sin(x - d)$ or $y = \cos(x - d)$ moves the graph d units right if d is positive, left if negative.

• Phase Shift: $y = \sin(x - d)$ is shifted d units right.

Vertical Translations

Adding a constant c to the function moves the graph vertically. $y = c + f(x)$ shifts the graph c units up if c is positive, down if negative.

• Example: $y = 3 - 2 \cos 3x$ is $y = -2 \cos 3x$ shifted up by 3 units.

General Form and Guidelines for Graphing

General Sinusoidal Function

The most general form is $y = c + a \sin[b(x - d)]$ or $y = c + a \cos[b(x - d)]$, where:

• a: Amplitude

• b: Frequency (affects period)

• c: Vertical shift

• d: Phase shift

Guidelines:

1. Find the interval for one period by solving $0 < b(x - d) < 2\pi$

2. Divide the interval into four equal parts

3. Evaluate the function at five key points

4. Plot and connect the points

5. Extend the graph as needed

Example: Graphing y = -1 + 2 sin(4x + π)

To graph $y = -1 + 2 \sin(4x + \pi)$:

• Express in standard form: $y = c + a \sin[b(x - d)]$

• Find period: $\frac{2\pi}{4} = \frac{\pi}{2}$

• Divide interval into four parts

• Calculate key points

Applications: Modeling with Sine Functions

Modeling Temperature Data

Sine functions can model real-world periodic phenomena, such as temperature variations.

• Example: Maximum monthly average temperature in New Orleans is 82°F, minimum is 54°F.

• Amplitude: $a = \frac{82 - 54}{2} = 14$

• Vertical shift: $c = \frac{82 + 54}{2} = 68$

• Period: 12 months, so $b = \frac{2\pi}{12} = \frac{\pi}{6}$

• Phase shift: Determined by data, trial and error gives $d = 4.2$

Summary Table: Key Points for Sinusoidal Functions

Tables are useful for organizing key values when graphing transformed sine and cosine functions.

x	Function Value	Transformed Value
0	sin(0)	0
$\frac{\pi}{2}$	sin($\frac{\pi}{2}$)	1
$\pi$	sin($\pi$)	0
$\frac{3\pi}{2}$	sin($\frac{3\pi}{2}$)	-1
$2\pi$	sin($2\pi$)	0

Additional info: Tables for more complex transformations (e.g., $y = -2 \sin 3x$, $y = 3 - 2 \cos 3x$, $y = -1 + 2 \sin(4x + \pi)$) are included in the images above.