Wave Navigator Notes Handout Wave Navigator
Charting the Course of Sine and Cosine Graphs
Subject: Algebra 2
Unit: Trig Functions
Student Name:
Date:
Period:
Part 1: The Periodic Map
A periodic function is a function that repeats its values in regular intervals or cycles.
Amplitude (\(a\))
Half the distance between the maximum and minimum values. It measures the height of the wave.
Period (\(P\))
The horizontal length of one complete cycle. For parent functions, this is \(2\pi\).
Midline (\(y = k\))
The horizontal line halfway between the maximum and minimum values. The wave oscillates around this line.
Phase Shift (\(h\))
The horizontal displacement of the graph from its parent position.
Part 2: The Parent Blueprints
\(y = \sin(x)\) Starts at Midline
\(x\) \(0\) \(\pi/2\) \(\pi\) \(3\pi/2\) \(2\pi\) \(y\)
\(\pi/2\) \(\pi\) \(3\pi/2\) \(2\pi\) \(1\) \(-1\)
\(y = \cos(x)\) Starts at Maximum
\(x\) \(0\) \(\pi/2\) \(\pi\) \(3\pi/2\) \(2\pi\) \(y\)
\(\pi/2\) \(\pi\) \(3\pi/2\) \(2\pi\) \(1\) \(-1\)
Part 3: Decoding the Equation
\(y = a \cdot \sin(b(x - h)) + k\)
\(|a|\)
Amplitude
\(2\pi/b\)
Period
\(h\)
Phase Shift
\(k\)
Midline (\(y=k\))
Graphing Checklist:
Identify \(a, b, h,\) and \(k\).
Calculate the Period using \(2\pi/b\).
Determine the 5 Key Points by dividing the period into 4 equal parts.
Sketch the Midline (\(y = k\)).
Apply shifts and plot the points. Connect with a smooth curve.
Navigational Check: Graph \(y = 3 \sin(2x) - 1\)
Amplitude:
Period:
Phase Shift:
Midline:
Calculations / Key Points:
Wave Navigator Slides Algebra 2
Wave Navigator
Graphing Sine & Cosine Functions
Trigonometry
Module 4: Periodic Models
The Periodic Path
A periodic function is a function that repeats its values in regular intervals or cycles.
Think: Seasons, tides, heartbeats, or a clock's hands.
Navigational Anatomy: Part 1
Amplitude (\(a\))
The vertical distance from the midline to a maximum or minimum value. It tells us how "tall" the wave is.
Midline (\(y=k\))
The average height of the graph. The line that cuts the graph perfectly in half horizontally.
Navigational Anatomy: Part 2
Period (\(P\))
The horizontal length of one full cycle. How far you travel along the x-axis before the pattern repeats.
Phase Shift (\(h\))
The horizontal shift. Sliding the wave left or right without changing its shape.
Blueprint: \(y = \sin(x)\)
Starts at Origin/Midline
\(x\) \(y\) \(0\) \(0\) \(\pi/2\) \(1\) \(\pi\) \(0\) \(3\pi/2\) \(-1\) \(2\pi\) \(0\)
Blueprint: \(y = \cos(x)\)
Starts at Maximum
\(x\) \(y\) \(0\) \(1\) \(\pi/2\) \(0\) \(\pi\) \(-1\) \(3\pi/2\) \(0\) \(2\pi\) \(1\)
The Navigation Console
\(y = \color{#06b6d4}a \color{white}\cdot \sin(\color{#06b6d4}b\color{white}(x - \color{#06b6d4}h\color{white})) + \color{#06b6d4}k\)
\(|a|\)
Amplitude
\(2\pi/b\)
Period
\(h\)
Phase Shift
\(k\)
Midline
The 5-Step Sketch
1
Extract Constants
Identify \(a, b, h, k\)
2
Calculate Period
\(P = 2\pi/b\)
3
Find Key Intervals
Divide \(P\) by 4
Wave Navigator Answer Key Answer Key
Wave Navigator: Charting the Course of Sine and Cosine Graphs
Subject: Algebra 2
Teacher Use Only
Part 1: The Periodic Map
A periodic function is a function that repeats its values in regular intervals or cycles.
Amplitude (\(a\))
Half the distance between the maximum and minimum values. It measures the height of the wave.
Period (\(P\))
The horizontal length of one complete cycle. For parent functions, this is \(2\pi\).
Midline (\(y = k\))
The horizontal line halfway between the maximum and minimum values. The wave oscillates around this line.
Phase Shift (\(h\))
The horizontal displacement of the graph from its parent position.
Part 2: The Parent Blueprints
\(y = \sin(x)\) Starts at Midline
\(x\) \(0\) \(\pi/2\) \(\pi\) \(3\pi/2\) \(2\pi\) \(y\) \(0\) \(1\) \(0\) \(-1\) \(0\)
\(y = \cos(x)\) Starts at Maximum
\(x\) \(0\) \(\pi/2\) \(\pi\) \(3\pi/2\) \(2\pi\) \(y\) \(1\) \(0\) \(-1\) \(0\) \(1\)
Part 3: Decoding the Equation
\(y = a \cdot \sin(b(x - h)) + k\)
\(|a|\)
Amplitude
\(2\pi/b\)
Period
\(h\)
Phase Shift
\(k\)
Midline (\(y=k\))
Graphing Checklist:
Identify \(a, b, h,\) and \(k\).
Calculate the Period using \(2\pi/b\).
Determine the 5 Key Points by dividing the period into 4 equal parts.
Sketch the Midline (\(y = k\)).
Apply shifts and plot the points. Connect with a smooth curve.
Navigational Check: Graph \(y = 3 \sin(2x) - 1\)
Amplitude: \(|3| = 3\)
Period: \(2\pi/2 = \pi\)
Phase Shift: \(0\)
Midline: \(y = -1\)
Calculations / Key Points:
Interval: \(\pi/4\)
Wave Navigator Exit Ticket Wave Navigator Exit Ticket
Mission: Terminal Check
Name
Date
Equation Analysis
\(y = 4\cos(x) + 2\)
Amplitude:
Period:
Midline:
Conceptual Check
Where does the parent function \(y = \sin(x)\) begin its cycle on the y-axis?
Sketch Transformation
Sketch one full cycle of \(y = 2\sin(x) - 1\)
0 \(\pi/2\) \(\pi\) \(3\pi/2\) \(2\pi\)
Wave Navigator Exit Ticket
Mission: Terminal Check
Name
Date
Equation Analysis
\(y = 4\cos(x) + 2\)
Amplitude:
Period:
Midline:
Conceptual Check
Where does the parent function \(y = \sin(x)\) begin its cycle on the y-axis?
Sketch Transformation
Sketch one full cycle of \(y = 2\sin(x) - 1\)
0 \(\pi/2\) \(\pi\) \(3\pi/2\) \(2\pi\)
