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Flip to reveal answersWrite the general exponential model and name each parameter.
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All 16 Flashcards — Exponential models
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Question
Write the general exponential model and name each parameter.
Answer
y = a · bˣ. a = initial value (y-intercept at x = 0). b = growth/decay factor per unit of x.
Question
In y = 500 · 1.06ˣ, interpret 500 and 1.06.
Answer
500 = initial value (at x = 0). 1.06 = growth factor — 6% growth per unit of x.
Question
If b > 1 in y = a · bˣ, is it growth or decay?
Answer
Growth — the output increases as x increases. The greater b is above 1, the faster the growth.
Question
If 0 < b < 1 in y = a · bˣ, is it growth or decay?
Answer
Decay — the output decreases as x increases. The closer b is to 0, the faster the decay.
Question
Population starts at 4000 and grows by 5% per year. Write the model.
Answer
P(t) = 4000 · 1.05ᵗ. Initial value a = 4000, growth factor b = 1 + 0.05 = 1.05.
Question
A substance starts at 200 g and halves every year. Write the model.
Answer
Q(t) = 200 · 0.5ᵗ. Initial value a = 200, decay factor b = 0.5.
Question
IB gives two data points for y = a · bˣ. How do you find a and b?
Answer
Substitute both points to get two equations. Divide one by the other to eliminate a and solve for b. Then substitute b back to find a.
Question
P = 3000 · 1.04ᵗ. Find P when t = 5.
Answer
P = 3000 · 1.04⁵ = 3000 · 1.2167 ≈ 3650.
Question
A student writes y = 5 · 1.03 · x instead of y = 5 · 1.03ˣ. What is the mistake?
Answer
y = 5 · 1.03 · x is linear, not exponential. In an exponential model, x must be the exponent: y = 5 · 1.03ˣ.
Question
Growth rate is 8%. A student writes b = 8. What is the correct value of b?
Answer
b is the growth factor, not the rate. b = 1 + rate = 1 + 0.08 = 1.08. Using b = 8 would give wildly wrong values.
Question
Can an exponential model y = a · bˣ ever give a negative value (with a > 0, b > 0)?
Answer
No — a · bˣ is always positive when a > 0 and b > 0. A negative result always means a calculation error.
Question
IB gives a table of data. How do you check if an exponential model fits?
Answer
Check the ratio of successive y-values: if y₂/y₁ is approximately constant, the data is exponential.
Question
What is the horizontal asymptote of y = 3 · 2ˣ? Explain.
Answer
y = 0. As x → −∞, 2ˣ → 0, so the whole expression approaches 0 from above. The x-axis is the asymptote.
Question
P(t) = 1000 · 0.8ᵗ. What happens to P as t → ∞?
Answer
P → 0. The substance/quantity decays toward zero but never fully disappears (according to the model).
Question
IB asks "Write down the equation of the horizontal asymptote" for y = 500 · 1.1ˣ.
Answer
y = 0. Write as a full equation. The growth model approaches 0 as x → −∞.
Question
Why might an exponential decay model be unreliable for very large t?
Answer
The model predicts the quantity approaches zero but never reaches it. In reality, the quantity may reach zero (e.g. a substance fully decays). The model is a mathematical idealisation.
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