Graphing Sine and Cosine Functions: Attributes, Symmetry, and Transformations

Graphing Trigonometric Functions

Introduction to Sine and Cosine Graphs

Trigonometric functions, especially sine and cosine, are fundamental in Precalculus for modeling periodic phenomena. Their graphs exhibit unique patterns and symmetries that are essential for understanding transformations and applications in mathematics and science.

• Sine and cosine functions are periodic and oscillate between maximum and minimum values.

• They are used to model waves, circular motion, and other repeating patterns.

Even and Odd Functions

Definitions and Properties

Functions can be classified as even, odd, or neither based on their symmetry properties.

• Even Functions: Satisfy $f(-x) = f(x)$ for all $x$ in the domain. Their graphs are symmetrical about the y-axis.

• Odd Functions: Satisfy $f(-x) = -f(x)$ for all $x$ in the domain. Their graphs are symmetrical about the origin.

• If $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$, the function is neither even nor odd.

Examples:

• Even: $f(x) = x^2$ $f(1) = (1)^2 = 1$ $f(-1) = (-1)^2 = 1$

• Odd: $f(x) = x^3$ $f(1) = (1)^3 = 1$ $f(-1) = (-1)^3 = -1$

Trigonometric Examples:

• $f(x) = \cos x$ is an even function because $\cos(-x) = \cos x$.

• $f(x) = \sin x$ is an odd function because $\sin(-x) = -\sin x$.

Attributes of Sine and Cosine Functions

Key Characteristics

The basic sine and cosine functions have several important attributes that define their graphs.

• Period: The length of one complete cycle. For $\sin x$ and $\cos x$, the period is $2\pi$.

• Domain: All real numbers, $(-\infty, \infty)$.

• Range: $[-1, 1]$ for the parent functions.

• Amplitude: The maximum distance from the midline to the peak, which is 1 for the parent functions.

• Midline: The horizontal axis that runs through the center of the graph, typically $y = 0$ for the parent functions.

• Intercepts:

  • Sine: y-intercept at (0, 0); x-intercepts at $x = 0, \pi, 2\pi, ...$

  • Cosine: y-intercept at (0, 1); x-intercepts at $x = \frac{\pi}{2}, \frac{3\pi}{2}, ...$

• Symmetry: Sine is odd (origin symmetry), cosine is even (y-axis symmetry).

Graphing Sine and Cosine: Table of Values

Using the Unit Circle

To graph $f(x) = \sin x$ and $f(x) = \cos x$, it is helpful to use key values from the unit circle.

x	-\frac{7\pi}{4}	-\frac{3\pi}{2}	-\frac{5\pi}{4}	-\pi	-\frac{3\pi}{4}	-\frac{\pi}{2}	-\frac{\pi}{4}	0	\frac{\pi}{4}	\frac{\pi}{2}	\frac{3\pi}{4}	\pi	\frac{5\pi}{4}	\frac{3\pi}{2}	\frac{7\pi}{4}
sin x	\frac{-\sqrt{2}}{2}	-1	\frac{-\sqrt{2}}{2}	0	\frac{\sqrt{2}}{2}	1	\frac{\sqrt{2}}{2}	0	\frac{\sqrt{2}}{2}	1	\frac{\sqrt{2}}{2}	0	\frac{-\sqrt{2}}{2}	-1	\frac{-\sqrt{2}}{2}
cos x	\frac{\sqrt{2}}{2}	0	\frac{-\sqrt{2}}{2}	-1	\frac{-\sqrt{2}}{2}	0	\frac{\sqrt{2}}{2}	1	\frac{\sqrt{2}}{2}	0	\frac{-\sqrt{2}}{2}	-1	\frac{-\sqrt{2}}{2}	0	\frac{\sqrt{2}}{2}

Additional info: The table above is inferred from the images and standard unit circle values for sine and cosine.

Graphing Sine and Cosine: Sketches and Features

Key Features of the Graphs

• Sine Graph: Starts at the origin (0,0), rises to 1 at $\frac{\pi}{2}$, returns to 0 at $\pi$, falls to -1 at $\frac{3\pi}{2}$, and completes the cycle at $2\pi$.

• Cosine Graph: Starts at (0,1), falls to 0 at $\frac{\pi}{2}$, to -1 at $\pi$, back to 0 at $\frac{3\pi}{2}$, and returns to 1 at $2\pi$.

• Periodicity: Both functions repeat every $2\pi$ units.

• Amplitude: The maximum value is 1, and the minimum is -1.

• Midline: The horizontal axis $y=0$ is the midline for both graphs.

Example: The graph of $y = \sin x$ passes through (0,0), $(\frac{\pi}{2}, 1)$, $(\pi, 0)$, $(\frac{3\pi}{2}, -1)$, and $(2\pi, 0)$.

Transformations of Sine and Cosine Functions

General Form and Effects

Transformations allow us to shift, stretch, compress, and reflect the basic sine and cosine graphs. The general form is:

$y = a \sin(b(x - h)) + k$ $y = a \cos(b(x - h)) + k$

• Amplitude (|a|): Vertical stretch/compression. The graph's maximum and minimum become $k + |a|$ and $k - |a|$.

• Period ($\frac{2\pi}{|b|}$): Horizontal stretch/compression. The period is shortened or lengthened by $b$.

• Phase Shift (h): Horizontal translation. The graph shifts left or right by $h$ units.

• Vertical Shift (k): Moves the graph up or down by $k$ units.

Example: $y = 2\sin(x - \frac{\pi}{4}) + 1$ has amplitude 2, period $2\pi$, phase shift $\frac{\pi}{4}$ to the right, and is shifted up by 1.

Developing Equations from Graphs

Identifying Parameters from a Graph

To write the equation of a sine or cosine function from its graph, identify the following:

• Amplitude: Measure the distance from the midline to a peak.

• Period: Find the horizontal length of one complete cycle.

• Phase Shift: Determine how far the graph is shifted horizontally from the standard position.

• Vertical Shift: Identify the midline's vertical position.

Example: If a sine graph has amplitude 3, period $\pi$, phase shift $\frac{\pi}{2}$ left, and midline at $y = -2$, its equation is $y = 3\sin(2(x + \frac{\pi}{2})) - 2$.

Summary Table: Sine vs. Cosine

Attribute	Sine ($\sin x$)	Cosine ($\cos x$)
Period	$2\pi$	$2\pi$
Amplitude	1	1
Domain	$(-\infty, \infty)$	$(-\infty, \infty)$
Range	$[-1, 1]$	$[-1, 1]$
y-intercept	0	1
x-intercepts	$0, \pi, 2\pi, ...$	$\frac{\pi}{2}, \frac{3\pi}{2}, ...$
Symmetry	Odd (origin)	Even (y-axis)

Additional info: Table values are standard for parent sine and cosine functions.