Periodic Pulse Worksheet Periodic Pulse
Deriving Equations from Graphs
Name:
Date:
Warm-up: The Mystery Wave
4 -2 2π
Analyze the graph on the left and determine the core attributes:
Midline (Equation):
Amplitude (|a|):
Video Insights: The Decision Process
Watch Example 3 (15:42 - 18:25). Focus on how the speaker decides between Sine and Cosine.
Why would you choose Sine vs. Cosine for the same graph?
Sine Shift Indicator
Look for where the graph crosses the:
Cosine Shift Indicator
Look for where the graph reaches a:
Practice: Sine or Cosine?
For each graph, identify the transformation variables and write TWO valid equations.
3 π
A: ______
B: ______
C: ______
D: ______
Sine Equation:
y =
Cosine Equation:
y =
1 π
A: ______
B: ______
C: ______
D: ______
Sine Equation:
y =
Cosine Equation:
y =
The Extension: The \(\pi/2\) Connection
Recall that \(\sin(x + \pi/2) = \cos(x)\). Look at your pairs of equations above. Explain how the period of the function (determined by \(b\)) changes the "standard" \(\pi/2\) shift needed to convert between sine and cosine.
Wave Whisperer Cards Wave Whisperer Cards
Strategic Discussion Prompts for Periodic Functions
STRATEGY 01
The Y-Axis Anchor
Look closely at the y-axis (\(x = 0\)).
Is the graph at the midline, a maximum, or a minimum?
"If it's at a peak, Cosine is your fastest route. If it's crossing the midline, Sine is your best friend."
STRATEGY 02
The Mirror Test
Instead of a phase shift, could a reflection (\(a < 0\)) make your equation simpler?
"Compare \(y = -\cos(x)\) to \(y = \cos(x - \pi)\). Which one would you rather write on a test?"
STRATEGY 03
The Shift Struggle
When you calculate the shift \(c\), you must look at the closest point of interest to the y-axis.
"Is it better to shift \(\pi/4\) to the left or \(7\pi/4\) to the right? Does the equation change?"
DISCUSSION
Equation Efficiency
A graph can be represented by infinite equations.
What makes one equation "better" than another in a real-world scenario?
Simplicity
Context
No Shifts
Aesthetics
Graph 1: Deep Dive
Compare your Sine and Cosine equations for Graph 1.
Which equation has the smaller phase shift?
Does the "direction" of the shift change your \(+/- sign\) inside the parentheses?
How did the midline change your \(d\) value?
Graph 2: Deep Dive
This graph completes two cycles in \(2\pi\).
How does the \(b=2\) affect the "distance" between the Sine and Cosine starting points?
If the period is shorter (\(\pi\)), is the \(\pi/2\) shift still the standard conversion?
Periodic Pulse Answer Key Answer Key
Periodic Pulse: Deriving Equations from Graphs
Warm-up: The Mystery Wave
Midline (d): \(y = 1\)
Calculation: \(\frac{4 + (-2)}{2} = \frac{2}{2} = 1\)
Amplitude (a): \(3\)
Calculation: \(4 - 1 = 3\) or \(1 - (-2) = 3\)
Video Insights: The Choice
Why choose Sine vs. Cosine? Both functions are horizontal shifts of each other (\(\pi/2\) phase difference). One might be easier because it requires a smaller or zero phase shift depending on the y-intercept.
Sine Shift Indicator
Crosses the midline at the start of a cycle.
Cosine Shift Indicator
Reaches a maximum or minimum at the start of a cycle.
Practice: Sine or Cosine?
Graph 1 (Midline \(y=0\), Amp 3, Period \(2\pi\), Shift Left \(\pi/4\))
Sine Equation
\(y = 3\sin(x + \pi/4)\)
Cosine Equation
\(y = 3\cos(x - \pi/4)\)
Variables: a=3, b=1, c=-\(\pi/4\) (for sine), d=0
Graph 2 (Midline \(y=0\), Amp 1, Period \(\pi\), No Shift for neg sine)
Sine Equation
\(y = -\sin(2x)\) or \(y = \sin(2(x - \pi/2))\)
Cosine Equation
\(y = \cos(2(x - \pi/4))\) or \(y = -\cos(2(x + \pi/4))\)
Variables: a=1, b=2, d=0
The Extension: The \(\pi/2\) Connection
The "standard" shift to convert sine to cosine is \(\pi/2\) when the period is \(2\pi\) (\(b=1\)).
Key Takeaway:
The phase shift required to convert between the two functions is always one-quarter of the period. Mathematically, if the conversion shift is \(S\), then \(S = \frac{Period}{4} = \frac{2\pi}{4b} = \frac{\pi}{2b}\).
Periodic Pulse Slides Periodic Pulse
Deriving Equations from Periodic Graphs
Warm-up: The Mystery Wave
Identify:
Midline (\(d\))
Amplitude (\(|a|\))
Mastering the Decision
Embedded media
"Why choose Sine over Cosine?"
— Focus on Example 3 (15:42-18:25)
Sine or Cosine?
A graph can be written using either function.
The Strategic Choice:
Smallest phase shift (\(c\))
Avoiding reflections if possible
Starting at key points (\(x=0\))
The Challenge
"Find TWO equations for every graph. One sine, one cosine. Be ready to defend which one is 'easier' to build."
THE \(\pi/2\) CONNECTION
\(\sin(x + \pi/2) = \cos(x)\)
If you shift a Sine wave left by 1/4 of its period, you get a Cosine wave.
Question for the Class:
If the period is \(\pi\), how much do we shift to convert?
