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All 16 Flashcards — Power and variation models
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Question
Write the general form of a power model.
Answer
y = axⁿ, where a is a constant and n is any real-number power.
Question
Give two real-world examples of power models.
Answer
Area of circle: A = πr² (power 2). Distance under gravity: s = 5t² (power 2). Surface area ∝ length² for similar shapes.
Question
In y = axⁿ, what is the key structural difference from an exponential model y = a · bˣ?
Answer
Power model: x is the base, n is a fixed exponent. Exponential: x is the exponent, b is a fixed base. Very different shapes for large x.
Question
In y = 3x², what happens to y when x doubles?
Answer
y increases by a factor of 2² = 4. Power models scale multiplicatively: doubling x multiplies y by 2ⁿ.
Question
y = 2x³ vs y = 2 · 3ˣ. Which is a power model and which is exponential?
Answer
y = 2x³ is a power model — x is the base. y = 2 · 3ˣ is exponential — x is the exponent.
Question
For large x, which grows faster — a power model or an exponential (b > 1)?
Answer
Exponential always eventually grows faster than any power model. Even x¹⁰⁰ is eventually overtaken by 2ˣ.
Question
A power model y = axⁿ with n > 0 passes through the origin. Does an exponential model?
Answer
No — exponential y = a · bˣ passes through (0, a), not the origin (unless a = 0). A power model with n > 0 passes through (0, 0).
Question
IB asks you to identify whether a model is power or exponential. You see y = 4 · 0.7ˣ. What is it?
Answer
Exponential — x is in the exponent. Base 0.7 means decay. It is NOT a power model.
Question
Which GDC regression type do you use for a power model?
Answer
Power regression (PwrReg on TI-84). Returns a and b for y = axᵇ.
Question
GDC gives PwrReg: a = 3.2456, b = 0.8123. How do you write the model?
Answer
y = 3.25x^0.812 (all values to 3 s.f.).
Question
When should you choose power regression over linear regression?
Answer
When the scatter plot shows a curved relationship (not straight), the data passes near the origin, and a straight line clearly doesn't fit the pattern.
Question
Power regression gives R² = 0.97. What does this tell you?
Answer
Very strong fit — 97% of variation in y is explained by the power model. It is a very good fit for the data.
Question
y = 0.5d^2.1 gives mass M (kg) vs diameter d (cm). What does the power 2.1 tell you?
Answer
Mass grows slightly faster than the square of diameter. Doubling d multiplies M by 2^2.1 ≈ 4.3.
Question
IB asks "Explain why this model may not be reliable for large x." How do you answer?
Answer
The model was built from data in a limited range. Using it for x well beyond that range is extrapolation — the pattern may not continue and the model may give unrealistic values.
Question
y = 2x^1.5. Find y when x = 4.
Answer
y = 2 · 4^1.5 = 2 · 8 = 16.
Question
A power model gives a negative y for a quantity that must be positive. What does this indicate?
Answer
The model is not valid for that input. Negative length, mass, or similar quantities are physically impossible. Either the input is outside the valid domain or the model breaks down.
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