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

Graphing Trigonometric Functions

Introduction to Sine and Cosine Graphs

Trigonometric functions such as sine and cosine are fundamental in Precalculus, especially for understanding periodic phenomena. Their graphs exhibit unique patterns and symmetries that are essential for modeling real-world cycles.

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

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

• Applications: Used in physics, engineering, and signal processing to model waves and oscillations.

Even and Odd Functions

Definitions and Properties

Functions can be classified based on their symmetry. Understanding whether a function is even, odd, or neither helps in graphing and analyzing their behavior.

Type	Definition	Symmetry	Example
Even	$f(-x) = f(x)$	Symmetrical about the y-axis	$f(x) = x^2$
Odd	$f(-x) = -f(x)$	Symmetrical about the origin	$f(x) = x^3$
Neither	$f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$	No symmetry	Various functions

• Cosine Function: $\cos(-x) = \cos(x)$, so cosine is even.

• Sine Function: $\sin(-x) = -\sin(x)$, so sine is odd.

Attributes of Sine and Cosine Functions

Key Characteristics

The graphs of sine and cosine share several attributes, including amplitude, period, domain, and range.

• Amplitude: The maximum distance from the midline to the peak. For $y = \sin x$ and $y = \cos x$, amplitude is 1.

• Period: The length of one complete cycle. For both functions, period is $2\pi$.

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

• Range: $[-1, 1]$.

• Midline: The horizontal axis about which the function oscillates, typically $y = 0$.

• Intercepts:

  • Sine: y-intercept at (0, 0); x-intercepts at $x = n\pi$, where $n$ is an integer.

  • Cosine: y-intercept at (0, 1); x-intercepts at $x = (2n+1)\frac{\pi}{2}$, where $n$ is an integer.

Graphing Activity: Sine and Cosine Values

Using the Unit Circle

To graph $f(x) = \sin x$ and $f(x) = \cos x$, use the unit circle to find function values at key points.

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	$\frac{\sqrt{2}}{2}$

Note: $\frac{\sqrt{2}}{2} \approx 0.71$

Sketching Sine and Cosine Graphs

Graph Features and Intercepts

Both sine and cosine graphs are smooth, continuous, and periodic. They oscillate between -1 and 1, with key intercepts and symmetry properties.

• Sine Graph: Starts at the origin (0,0), peaks at $\frac{\pi}{2}$, returns to zero at $\pi$, trough at $\frac{3\pi}{2}$, and completes the cycle at $2\pi$.

• Cosine Graph: Starts at (0,1), drops to zero at $\frac{\pi}{2}$, trough at $\pi$, returns to zero at $\frac{3\pi}{2}$, and peaks again at $2\pi$.

• Midline: The horizontal axis $y=0$ divides the graph into equal parts vertically.

Transformations of Sine and Cosine Functions

General Form and Effects

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

• General Equation: $y = A \sin(B(x - C)) + D$ or $y = A \cos(B(x - C)) + D$

• Amplitude ($A$): Vertical stretch/compression. $|A|$ is the new amplitude.

• Period ($B$): $\text{Period} = \frac{2\pi}{|B|}$

• Phase Shift ($C$): Horizontal shift by $C$ units.

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

Example: $y = 4(x - 3)^2 + 4$ is a transformed quadratic, but for sine/cosine, $y = 2\sin(x - \frac{\pi}{4}) + 1$ would have amplitude 2, phase shift $\frac{\pi}{4}$, and vertical shift 1.

Developing Equations from Graphs

Identifying Parameters from a Graph

To write the equation of a sine or cosine graph, identify amplitude, period, phase shift, and vertical shift from the graph's features.

• Step 1: Find the midline (vertical shift $D$).

• Step 2: Measure the distance from midline to peak (amplitude $A$).

• Step 3: Determine the period (distance for one full cycle).

• Step 4: Locate the phase shift ($C$) by identifying where the cycle starts.

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

Summary Table: Sine vs. Cosine Function Properties

Property	Sine ($\sin x$)	Cosine ($\cos x$)
Amplitude	1	1
Period	$2\pi$	$2\pi$
Domain	$(-\infty, \infty)$	$(-\infty, \infty)$
Range	$[-1, 1]$	$[-1, 1]$
Symmetry	Odd	Even
y-intercept	0	1
x-intercepts	$x = n\pi$	$x = (2n+1)\frac{\pi}{2}$

Additional info: The notes also briefly mention transformations of other parent functions (e.g., $y = x$, $y = x^2$), but the main focus is on sine and cosine. The quadratic example $y = 4(x-3)^2 + 4$ is included to illustrate general transformation principles.