The Wave Builder
Unit 8: Graphs of Trigonometric Functions
Step 1 of 9: Amplitude
A sound wave at normal volume. Drag the volume slider and watch what happens to the wave. What changes? What stays the same?
Step 2 of 9: Predict the Maximum
Set amplitude to 3. What will the maximum value be? The minimum?
Step 3 of 9: Negative Amplitude
What does \(A = -2\) do to the curve? Pick your prediction, then see.
Step 4 of 9: Period
A musical note. Raise the pitch by dragging the B slider. The wave compresses. More cycles fit in the same space.
Step 5 of 9: Predict the Period
\(B = 2\). What is the period?
Step 6 of 9: Vertical Shift
A city's temperature over a year: summer high 95, winter low 45. What is the midline, the average temperature?
Step 7 of 9: Phase Shift
Two cities with the same temperature pattern, but one is in the Southern Hemisphere -- seasons offset by six months. Drag C to slide the curve horizontally.
Step 8 of 9: The Sign Trap
Look at the equation \(y = \sin\!\left(x - \frac{\pi}{4}\right)\). Which direction does the curve shift?
Step 9 of 9: Full Assembly
Build this: amplitude 3, period \(\pi\), shifted \(\frac{\pi}{4}\) right, midline \(y = 2\). Set all four sliders.
Tolerances: A within 0.5, B within 0.25, C within π/12, D within 0.5.
Explore the Wave Equation
Adjust the sliders to transform the wave. The equation updates live in two forms so you can see why factoring matters.
Try This
Model a Ferris Wheel
A Ferris wheel has a 60-foot diameter. Its center is 35 feet above the ground. It completes one rotation every 4 minutes. You board at the lowest point.
Find A, B, C, and D. Should you use sine or cosine, and why does boarding at the bottom point you toward negative cosine?
Tolerances: A within 2, B within 0.2, C within 0.5, D within 2. Use \(\pi/2 \approx 1.57\) for B.
Blood Pressure Monitor
A blood pressure monitor reads 120 over 80 with a heart rate of 72 bpm. Write a sinusoidal model for pressure over time in seconds.
What is the amplitude? The midline? The period?
Tolerances: amplitude within 2, midline within 2, period within 0.05 seconds.
Match the Wave
A target curve appears. Set the four sliders until your curve lands on top of it. Three difficulty levels control how much help you get.
- Identify how amplitude (A) affects the height of a sinusoidal graph
- Calculate period from the frequency parameter B using \(\text{Period} = \frac{2\pi}{B}\)
- Determine the direction and magnitude of horizontal (phase) shifts
- Model real-world periodic phenomena using \(y = A\sin(B(x - C)) + D\)
Quick Check
For \(y = 3\sin\!\left(2\!\left(x - \frac{\pi}{6}\right)\right) + 1\), what is the period and vertical shift?
Instructor Notes
Teaching Notes
- Start with the audio analogy. Students who have used equalizers understand amplitude and frequency before they have the math vocabulary.
- The phase shift sign confusion (Step 8) is the most common exam error in this unit. Spend time here.
- Show both equation forms side by side: \(y = A\sin(Bx - C)\) vs \(y = A\sin(B(x - C))\). The first is what textbooks print. The second is what reveals the actual shift.
- Use the Ferris wheel problem to connect all four parameters to one physical object students can visualize.
Common Student Errors
- Reading B as the period instead of computing \(2\pi/B\)
- Getting phase shift backwards when \(B \neq 1\) and the equation is not factored
- Confusing amplitude with the distance from peak to trough (amplitude is half that)
- Forgetting that negative A reflects, not shrinks
Discussion Questions
- Why does a larger B produce a shorter period? Connect to the audio analogy.
- If you know a real-world cycle repeats every 365 days, how do you find B?
- Why does negative cosine model a Ferris wheel when you board at the bottom?
Exam Connection
- "Given \(y = -2\sin(3x - \pi/2) + 4\), identify amplitude, period, phase shift, and midline" is a standard exam item. Factor B first to read the shift.
- Graphing from equation: students must plot at least one full period with five key points labeled.