Honors Algebra II / Precalculus

Wave Riders: Guided Notes

Anatomy & Parameters of Transformed Trigonometric Waves

Name: ______________________ Date: _________

Class: _________________

Standard Equations of Trigonometric Waves

Ocean waves, light, sound, tides, and Ferris wheels all exhibit periodic behavior. We model these repeating patterns by transforming the parent sine and cosine functions.

Sine Wave model

y = a • sin(b(x − h)) + k

Cosine Wave model

y = a • cos(b(x − h)) + k

Key Variables & Mathematical Formulas

Complete the definitions and formulas below to unlock the wave transformation toolkit:

Letter & Variable	What it Does (Physical Effect)	Calculation & Formula Rule
a (Amplitude)	Controls the vertical stretch/compression. Wave height from the ________________.	a = max − min __________ (Always positive distance value)
b (Frequency)	Controls horizontal stretch/compression. Number of cycles in a __________ interval.	b = __________ Period
Period	The length of __________ complete cycle. How long it takes the wave to repeat.	Period = 2π __________
h (Phase Shift)	Controls horizontal translation. Shift Left: (x + h) \| Shift __________: (x − h)	Initial start point of the pattern cycle.
k (Vertical Shift)	Controls vertical translation. Defines the center/________________ of the wave.	k = max + min __________ Midline line equation: y = k

Visual Anatomy of a Sine Wave

Label the diagram below using the keywords: Amplitude, Midline (y=k), Period, and Phase Shift (h).

① _______________________

② _______________________

③ _______________________

④ _______________________

Part II: Guided Practice

Graphing from Wave Equations

Drafting wave structures systematically using the Five-Point Method

Wave Riders Unit Topic 5.2

1 Vertical Bounds

Sketch the midline y = k. Mark the boundaries at y = k ± a.

2 Cycle Range

Start at shift h. Add the Period to locate cycle end.

3 The 5 Key Points

Divide period into 4 quarters. Plot start, mid, end, max/min.

Guided Walkthrough: Graphing y = 2·sin(2(x − π/4)) + 1

Sine Wave

Calculate Parameters:

Amplitude a = _______

Frequency b = _______

Period =

2π b

= _______

Phase Shift h = _______ (________)

Vertical Shift k = _______

Critical coordinates:

Midline: y = _______

Maximum: _______ | Minimum: _______

Horizontal bounds of wave:

Starts at _______ | Ends at _______

Quarter Width =

Period 4

= _______

Point Position	Start (x = h)	1st Quarter	Midpoint	3rd Quarter	End (x = h + Pd)
x-value	__________	__________	__________	__________	__________
Pattern Behavior	Midline	Maximum	Midline	Minimum	Midline
y-value	__________	__________	__________	__________	__________

Grid Setup & Wave Sketch

Step 1: Sketch midline. Step 2: Scale and label the axes. Step 3: Plot the 5 key points and connect with a smooth, curved wave.

Part III: Applications

Writing Equations & Real-World Modeling

Extracting trigonometric variables from mathematical graphs and real-life rhythms

Wave Riders Unit Topic 5.3

Section 1: Working Backwards (From Graph to Equation)

Given Graph:

Wave cycle from x = 0 to π

Follow the formulas to build the cosine/sine equation:

1. Find Peak and Trough:

Max = _______ | Min = _______

2. Midline (Vertical Shift, k):

k =

Max + Min 2

= _______

3. Amplitude (a):

a =

Max − Min 2

= _______

4. Period & Frequency (b):

Cycle length Period = _______ | b =

2π Period

= _______

Completed Sine Model (with h = 0):

y = ________________________________________

Section 2: Ocean Harbor Tides Scenario

The Pier Deep-Water Dock: The ocean water depth fluctuates periodically. At high tide (midnight, t = 0), the depth reaches a maximum of 16 meters. Low tide occurs exactly 6 hours later, at a minimum depth of 4 meters.

1. Identify Key Details:

Maximum Depth = ______ m
Minimum Depth = ______ m

2. Find the Midline (k):

k =

Max + Min 2

= ______ m
(This is average water depth)

3. Find the Amplitude (a):

a =

Max − Min 2

= ______ m
(Max deviation from average)

4. Find the Period & Frequency (b):

Time from High to Low = 6 hours.
Full cycle Period = ______ hours
Calculate b =

2π Period

= ______

5. Formulate the Cosine Function Model

Since the depth cycle starts at high tide (maximum height) at t = 0, we model this using a standard positive cosine function with phase shift h = 0: d(t) = a · cos(b · t) + k.

d(t) = _________________________________________

Wave Riders Guided Notes

Wave Riders Guided Notes

Similar Materials

Math Educator Candidacy Portfolio

Kayla Borozny

Step Strategy Teacher Key

tafranklin

Step Strategy Student Assessment

tafranklin

Step Builders Assessment Answer Key

tafranklin

Step Builders Assessment Worksheet

tafranklin

Clue Cards Desk Guide

tafranklin

Error Tracker Teacher Rubric

tafranklin

Decoder Clues Assessment

tafranklin

Winning Formula Worksheet

Karen Dhuyvetter

Draft Day Activity

Karen Dhuyvetter

Front Office Slides

Karen Dhuyvetter

Front Office Lesson Plan

Karen Dhuyvetter

Candy Shifters Teacher Guide

Julia Butz

Candy Shifters Worksheet

Julia Butz

Candy Shifters Slides

Julia Butz

Practice Sheet Answer Key

Aysenur Ersoy

Bakery Blitz Level Up Answer Key

Danielle Gagnon

Bakery Blitz Level Up Worksheet

Danielle Gagnon

Power Multiply Answer Key

Alyssa Smith

Power Multiply Worksheet

Alyssa Smith

Wave Riders Teacher Key

Aysenur Ersoy

Wave Riders Practice Worksheet

Aysenur Ersoy