Concert Decibels Warmup Activity
Algebra II • Function Transformations
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Concert Decibel Dilemma: The Sound Engineer's Hook
Bellringer Warmup
The Real-World Problem
You are mixing sound at an outdoor festival across a 120-meter lawn. Volume peaks directly in front of the center stage speakers and falls off symmetrically as fans walk to the left or right. The sound level \( S(x) \) in decibels (dB) at position \( x \) meters across the lawn is modeled by:
\( S(x) = -1.5\,|\,x - 60\,| + 105 \)
Stage Decibel Profile 0m to 120m Lawn
Peak (60, 105 dB) 0m (Left) 60m (Center) 120m (Right) 105 dB 60 dB
Rate of change: \( \pm 1.5 \) dB per meter away from center stage
Audio Control Panel
\( h = 60 \): Stage location (Horizontal shift)
\( k = 105 \): Master volume (Vertical shift)
\( a = -1.5 \): Dropoff rate & inverted V
Standard Form: \( f(x) = a\,|\,x - h\,| + k \)
Sound Check Inquiry Prompts
Predict how sound board adjustments transform the function
1. Locate the "Sweet Spot"
Vertex
Where on the lawn is sound loudest, and what is the peak decibel level? Which graphical feature does this point represent?
2. Noise Ordinance (dB Drop)
Shift \( k \)
City rules require lowering peak volume to 95 dB without moving the speakers. Write the new equation. Which value changed?
3. Moving the Main Stage
Shift \( h \)
The stage shifts 20 meters right (to \( x = 80 \)). Inside the absolute value bars, will the term be \( |x - 80| \) or \( |x + 80| \)? Why?
4. Speaker Baffles (Spread)
Scale \( a \)
Diffusers spread the sound wider so it drops only 0.5 dB per meter. Does the graph get wider or narrower? What is the new value of \( a \)?
Lesson Big Idea: In \( f(x) = a\,|\,x - h\,| + k \), the vertex \( (h, k) \) anchors the sweet spot, while \( a \) controls spread and direction!
Vertex = (h, k)
