Amplitude, Period, and Phase Shift

General Sinusoidal Form

$y = A \sin(B(x - C)) + D$

$y = A \cos(B(x - C)) + D$

Amplitude ($|A|$)

The amplitude is half the distance between maximum and minimum values.

$\text{Amplitude} = |A| = \frac{\max - \min}{2}$

• $|A| > 1$: Vertical stretch
• $0 < |A| < 1$: Vertical compression
• $A < 0$: Reflection over x-axis (inverted)

Period

The period is the horizontal length of one complete cycle.

For Sine and Cosine

$\text{Period} = \frac{2\pi}{|B|}$

For Tangent and Cotangent

$\text{Period} = \frac{\pi}{|B|}$

Finding B from Period

$B = \frac{2\pi}{\text{Period}} \quad \text{(for sin, cos)}$

$B = \frac{\pi}{\text{Period}} \quad \text{(for tan, cot)}$

Phase Shift ($C$)

The phase shift is the horizontal displacement from the standard position.

$\text{Phase Shift} = C$

• $C > 0$: Shift RIGHT by $C$ units
• $C < 0$: Shift LEFT by $|C|$ units

Finding Phase Shift

From $y = A \sin(Bx - \varphi)$:

$y = A \sin(B(x - \varphi/B))$

$\text{Phase Shift} = \frac{\varphi}{B}$

Vertical Shift ($D$)

The vertical shift moves the midline up or down.

$\text{Midline: } y = D$

• $D > 0$: Shift UP by $D$ units
• $D < 0$: Shift DOWN by $|D|$ units

Finding Vertical Shift

$D = \frac{\max + \min}{2}$

Summary Table

Parameter	Effect	How to Find
$A$	Amplitude	$\frac{\max - \min}{2}$
$B$	Period	$\frac{2\pi}{\text{Period}}$
$C$	Phase shift	Horizontal displacement
$D$	Vertical shift	$\frac{\max + \min}{2}$

Examples

Example 1: $y = 3 \sin(2x)$

Parameter	Value
Amplitude	$3$
Period	$\frac{2\pi}{2} = \pi$
Phase Shift	$0$
Vertical Shift	$0$

Example 2: $y = -2 \cos(x - \frac{\pi}{4}) + 1$

Parameter	Value
Amplitude	$\lvert -2 \rvert = 2$
Period	$\frac{2\pi}{1} = 2\pi$
Phase Shift	$\frac{\pi}{4}$ (right)
Vertical Shift	$1$ (up)
Reflection	Yes (inverted)

Example 3: $y = 4 \sin(\frac{\pi x}{2} + \pi)$

Rewrite: $y = 4 \sin(\frac{\pi}{2}(x + 2))$

Parameter	Value
Amplitude	$4$
Period	$\frac{2\pi}{\pi/2} = 4$
Phase Shift	$-2$ (left 2)
Vertical Shift	$0$

Frequency

Frequency ($f$) is the number of cycles per unit:

$f = \frac{1}{\text{Period}} = \frac{|B|}{2\pi}$

Also:

$\text{Angular Frequency: } \omega = 2\pi f = |B|$

Writing Equations from Graphs

1. Identify the midline → $D$
2. Find amplitude → $|A|$
3. Determine if inverted → sign of $A$
4. Measure period → solve for $B$
5. Find phase shift → $C$

Standard vs. Shifted Forms

Form	Equation
Standard	$y = A \sin(Bx) + D$
Phase Shift	$y = A \sin(B(x - C)) + D$
Alternative	$y = A \sin(Bx - \varphi) + D$ where $C = \varphi/B$