section 4.4

Trigonometric Functions and Their Graphs

Introduction to Sine and Cosine Functions

Trigonometric functions such as sine and cosine are fundamental in Precalculus, describing periodic phenomena and circular motion. Their graphs exhibit regular patterns characterized by amplitude, period, and phase shift.

• Sine Function: $y = \sin(x)$ starts at the midline and oscillates between -1 and 1.

• Cosine Function: $y = \cos(x)$ starts at its maximum value and also oscillates between -1 and 1.

• Period: The time it takes to complete one full cycle. For $\sin(x)$ and $\cos(x)$, the period is $2\pi$.

• Amplitude: Half the distance between the maximum and minimum values. For $y = a\sin(x)$ or $y = a\cos(x)$, amplitude is $|a|$.

Example: The graph of $y = \sin(x)$ completes one cycle from $x = 0$ to $x = 2\pi$.

Transformations of Trigonometric Functions

General Form and Parameters

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

• $y = a \sin(b(x - c)) + d$

• $y = a \cos(b(x - c)) + d$

Parameter	Effect
$a$	Amplitude ($|a|$)
$b$	Period ($\frac{2\pi}{|b|}$)
$c$	Horizontal (phase) shift
$d$	Vertical shift (midline)

Example: $y = 2\sin(3(x - \frac{\pi}{4})) - 1$ has amplitude 2, period $\frac{2\pi}{3}$, phase shift $\frac{\pi}{4}$ to the right, and is shifted down by 1.

Graphing Sine and Cosine Functions

Steps for Graphing

To graph a transformed sine or cosine function, follow these steps:

1. Identify amplitude ($|a|$), period ($\frac{2\pi}{|b|}$), phase shift ($c$), and vertical shift ($d$).

2. Mark the midline at $y = d$.

3. Plot key points: start, maximum, minimum, and end of one period.

4. Apply the phase shift by moving the graph horizontally.

5. Draw the curve, ensuring the correct amplitude and period.

Example: Graph two periods of $y = \frac{1}{2}\sin(2x - \frac{\pi}{3})$:

• Amplitude: $\frac{1}{2}$

• Period: $\pi$

• Phase shift: $\frac{\pi}{6}$ to the right

• Vertical shift: none

Special Properties and Applications

Midline and Amplitude

The midline is the horizontal axis about which the function oscillates, given by $y = d$. The amplitude is the maximum deviation from the midline.

• Midline: $y = d$

• Amplitude: $|a|$

Example: For $y = 3\sin(x) + 2$, the midline is $y = 2$ and amplitude is 3.

Summary Table: Sine and Cosine Transformations

Function	Amplitude	Period	Phase Shift	Vertical Shift
$y = a\sin(b(x-c)) + d$	$|a|$	$\frac{2\pi}{|b|}$	$c$	$d$
$y = a\cos(b(x-c)) + d$	$|a|$	$\frac{2\pi}{|b|}$	$c$	$d$

Additional info:

• Graphs often start at the midline for sine and at the maximum for cosine.

• Negative amplitude reflects the graph over the midline.

• Transformations can be combined for more complex graphs.