Power model: y = axⁿ: A power model describes situations where one quantity is proportional to a power of another.
The exponent n determines the shape.
Unlike exponential models (where x is in the exponent), in a power model x is the base.
- Scaling constant (positive for real-world models)
- Exponent — can be any real number (2, 3, 0.5, −1, etc.)
- Independent variable (the base)
| n value | Shape | Real-world example |
|---|---|---|
| n = 2 | Parabola (upward) | Area of circle: A = πr² |
| n = 3 | Cubic | Mass ∝ (height)³ for animals (Rubner's law) |
| n = 0.5 | Square root curve | Speed of waves on water |
| n = −1 | Hyperbola (inverse proportion) | Smoothie sales: n = 40000/x² (n = −2) |
| n = −2 | Steeper hyperbola | Intensity ∝ 1/distance² |
Power-model shapes — how n changes the curve. y = x² is a parabola; y = x³ rises much faster and has reflective symmetry through the origin.
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Power model vs exponential model — easy to confuse: These two look similar but are fundamentally different.
In a power model the variable x is the BASE.
In an exponential model x is the EXPONENT.
| Model type | Form | x is... | Example |
|---|---|---|---|
| Power | y = axn | the BASE | Mass = 125 × h³ (h = height) |
| Exponential | y = a × bx | the EXPONENT | Population = 200 × 1.15t |
Big cats: mass m (kg) is proportional to (height h)³.
A cheetah has m = 64 kg at h = 0.8 m.
(a) Find the constant of proportionality.
(b) Find m when h = 0.75 m.
Step by step
- Write the power model.
- Substitute known values to find a.
- Write the complete model.
- Substitute h = 0.75.
Final answer
The constant is a = 125. A cheetah of height 0.75 m has mass ≈ 52.7 kg.
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GDC PwrReg finds the best-fit power model: When given a table of data that follows a power relationship, use the GDC's Power Regression (PwrReg) to find the values of a and n that best fit the data.
GDC steps for power regression (TI-84)
- Enter x-values in L1, y-values in L2.
- Press STAT → CALC → A:PwrReg.
- Confirm L1, L2. Press ENTER.
- Read off a and b (the GDC uses "b" for the exponent).
- Store to Y1 using RegEQ if you want to graph and evaluate.
A shop models smoothie sales as n = axb where x is price (pesos).
Data gives PwrReg: a = 40000, b = −2.
Find the maximum number of smoothies sold per day if each smoothie costs 50 pesos.
Step by step
- Write the model from GDC output.
- Substitute x = 50.
Final answer
At a price of 50 pesos, the shop sells 16 smoothies per day.
Check: does the model make physical sense?: After finding a power model, always verify it gives sensible values at the extremes.
If the model predicts a negative mass or infinite sales, something is wrong with the domain.
✗ Common mistakes
- Confusing a power model y = axn with y = ax (exponential)
- Forgetting units — "m = 52.7" means nothing without "kg"
- Not checking whether the GDC a and n match the physical story
✓ Correct approach
- Check: is x the base (power) or the exponent (exponential)?
- Always attach units to every answer: m ≈ 52.7 kg
- Verify with one known data point: does 125(0.8)³ = 64? Yes ✓
Direct vs inverse proportion: n > 0: as x increases, y increases (direct power proportion). n < 0: as x increases, y decreases (inverse power proportion).
State this in your interpretation: "As height increases, mass increases (direct proportion to h³)."