Graphing Sine and Cosine Functions: Properties, 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 used to model many real-world phenomena such as sound waves, tides, and circular motion. Understanding their properties and transformations is essential in Precalculus.

• Parent Functions: The basic forms are $y = \sin x$ and $y = \cos x$.

• Periodicity: Both functions repeat their values in regular intervals, known as the period.

• Applications: Used in physics, engineering, and signal processing.

Even and Odd Functions

Definitions and Symmetry

Functions can be classified as even, odd, or neither based on their symmetry properties. This classification helps in understanding the behavior of their graphs.

• 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.

• Neither: If a function does not satisfy either property, it 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 even because $\cos(-x) = \cos x$.

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

Attributes of Sine and Cosine Functions

Key Properties

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

• Period: The length of one complete cycle. For $y = \sin x$ and $y = \cos x$, 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 that runs through the center of the graph, typically $y = 0$.

• 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 to Find Key Points

To graph $y = \sin x$ and $y = \cos x$, use the unit circle to determine the function values at important angles.

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 Graphs

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 $2\pi$.

Key Features:

• Midline: $y = 0$

• Amplitude: 1

• Period: $2\pi$

• Range: $[-1, 1]$

• Domain: $(-\infty, \infty)$

• Symmetry: Sine is odd, cosine is even

Transformations of Sine and Cosine Functions

General Form and Effects

Transformations allow us to shift, stretch, compress, and reflect the parent graphs of sine and cosine functions. 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 changes from $2\pi$ to $\frac{2\pi}{|b|}$.

• Phase Shift ($h$): Horizontal translation. The graph shifts right if $h > 0$, left if $h < 0$.

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

Example: $y = 4(x - 3)^2 + 4$ is a transformation of the parent function $y = x^2$ (not a trig function, but illustrates the concept of shifting and stretching).

Summary Table: Sine and Cosine Function Properties

Property	$y = \sin x$	$y = \cos x$
Amplitude	1	1
Period	$2\pi$	$2\pi$
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	Even