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

Graphing Trigonometric Functions

Introduction to Sine and Cosine Graphs

The sine and cosine functions are fundamental periodic functions in trigonometry. Their graphs, called sinusoids, are essential for modeling periodic phenomena in mathematics, science, and engineering.

• Periodic Functions: Both sin(x) and cos(x) repeat their values in regular intervals, known as periods.

• Applications: Used to model waves, oscillations, and circular motion.

Even and Odd Functions

Definitions and Symmetry

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

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

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

• Neither: If a function does not satisfy either property, it is neither even nor odd.

Type	Definition	Example	Symmetry
Even	$f(-x) = f(x)$	$f(x) = x^2$	y-axis
Odd	$f(-x) = -f(x)$	$f(x) = x^3$	Origin

• Cosine Function: $f(x) = \cos x$ is an even function.

• Sine Function: $f(x) = \sin x$ is an odd function.

Attributes of Sine and Cosine Functions

Key Properties

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

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

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

• Range: $[-1, 1]$.

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

• Midline: The horizontal axis about which the function oscillates, 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 $x = \frac{\pi}{2}, \frac{3\pi}{2}, ...$

Graphing Activity: Sine and Cosine Values

Using the Unit Circle

Values of $\sin x$ and $\cos x$ at key angles can be found using the unit circle. These values are used to plot the graphs of the functions.

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

Additional info: $\frac{\sqrt{2}}{2} \approx 0.71$

Graphing the Sine and Cosine Functions

Sketching the Parent Functions

The graphs of $y = \sin x$ and $y = \cos x$ are smooth, continuous waves that repeat every $2\pi$ units along the x-axis.

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

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

Attributes Summary Table

Function	Period	Domain	Range	Even/Odd	y-intercept	x-intercepts
$\sin x$	$2\pi$	$(-\infty, \infty)$	$[-1, 1]$	Odd	0	$...,-2\pi,-\pi,0,\pi,2\pi,...$
$\cos x$	$2\pi$	$(-\infty, \infty)$	$[-1, 1]$	Even	1	$...,-3\pi/2,-\pi/2,\pi/2,3\pi/2,...$

Transformations of Sine and Cosine Functions

General Form and Effects

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

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

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

• Amplitude ($|a|$): Vertical stretch/compression. The graph oscillates between $k + a$ and $k - a$.

• Period ($\frac{2\pi}{|b|}$): Horizontal stretch/compression. The period changes based on $b$.

• 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: $y = 4(x - 3)^2 + 4$ is a vertically stretched and shifted parabola (not a trig function, but illustrates transformation concepts).

Additional info: For trigonometric functions, transformations are applied similarly to other parent functions, but the periodic nature must be considered.