Growth Factor Quiz: Interpreting Initial Value and Growth Factor

  • 9th Grade
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| Attempts: 12 | Questions: 20 | Updated: Dec 17, 2025
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1) In N(t) = a·b^t, the coefficient a equals the value at t = 0.

Explanation

Evaluate at t = 0: N(0) = a·b^0 = a. So a is the initial value.

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About This Quiz
Growth Factor Quiz: Interpreting Initial Value and Growth Factor - Quiz

How does a growth factor shape the behavior of an exponential model? In this quiz, you’ll explore how initial value and growth rate interact to create predictable long-term patterns. You’ll identify growth factors from equations, interpret how they affect changes between terms, and compare situations with faster or slower increases.... see moreThrough guided problems, you’ll build intuition for how exponential growth works, why small rate differences matter, and how to read meaning directly from the model’s structure.
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2) A culture has P(0) = 200 and P(1) = 260. Assuming P(t) = a·b^t, which pair (a,b) fits?

Explanation

From P(0) = a = 200. Then P(1) = a·b = 260 ⇒ 200·b = 260 ⇒ b = 1.3. So (a,b) = (200, 1.3).

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3) For b = 1.2 in N(t) = a·b^t, the exact doubling time (in periods) is:

Explanation

Doubling time T solves a·b^T = 2a ⇒ b^T = 2 ⇒ T = ln 2 / ln b = ln 2 / ln 1.2.

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4) If b = 1 in N(t) = a·b^t, then N(t) stays constant over time.

Explanation

With b = 1, N(t) = a·1^t = a for all t, so the value is constant.

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5) Which model matches “starts at 1000 and grows by 5% each hour”?

Explanation

A 5% hourly increase means multiply by 1.05 each hour: N(t) = 1000(1.05)^t.

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6) In N(t) = a·b^{t/3}, the multiplier over 3 time units is b.

Explanation

Increasing t by 3 multiplies the exponent t/3 by 1, so N(t+3) = a·b^{(t+3)/3} = a·b^{t/3}·b. Thus the factor over 3 units is b.

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7) If b = 0.8 in y = a·b^t, what is y after two periods relative to a?

Explanation

After 2 periods: y(2) = a·(0.8)^2 = a·0.64 = 0.64a.

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8) For N(t) = 300(1.25)^t, what does the coefficient 300 represent?

Explanation

In N(t) = a·b^t, a is the initial value because N(0) = a·b^0 = a. Here N(0) = 300.

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9) In N(t) = 300(1.25)^t, what does 1.25 represent?

Explanation

The base b = 1.25 is the multiplicative growth factor each period: multiply by 1.25 every step.

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10) For N(t) = a·b^t with b = 1.25, what is the percent rate of change per period?

Explanation

Percent rate per period = (b − 1)·100%. With b = 1.25, this is (1.25−1)·100% = 25%.

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11) Suppose P(t) = a·b^t and P(3) = 2a. What is b?

Explanation

P(3) = a·b^3 = 2a ⇒ b^3 = 2 ⇒ b = 2^(1/3).

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12) For N(t) = 120(1.5)^t, compute N(2).

Explanation

N(2) = 120·(1.5)^2 = 120·2.25 = 270.

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13) For P(t) = 1200(0.9)^t, the percent change per period is ____.

Explanation

Since b = 0.9, percent change = (0.9 − 1)·100% = −10%, meaning a 10% decrease each period.

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14) Which statements about M(t) = 500(1.03)^t are true? Select all that apply.

Explanation

Initial value a = 500. Base b = 1.03 is the per‑period multiplier, which equals a 3% increase since (1.03−1)·100% = 3%. It does not add 3 units; and M(1) = 500·1.03 = 515 is true.

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15) For S(t) = 800(0.75)^t, S(0) equals ____.

Explanation

At t = 0, S(0) = 800·(0.75)^0 = 800.

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16) For y(t) = a·b^t with a > 0 and 0 < b < 1, which are true? Select all that apply.

Explanation

All statements hold: b 0 the output stays positive, and multiplication by b happens each period.

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17) Given data follow an exponential model with P(0) = 50 and P(1) = 60. Which (a,b) fits?

Explanation

From P(0) = a = 50. Then b = P(1)/a = 60/50 = 1.2. So (50, 1.2).

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18) In N(t) = a·b^t, the percent change per period equals ____%.

Explanation

Percent change = (b − 1)·100%. For b > 1 this is positive growth; for 0

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19) Which pairs (a,b) represent exponential growth for y = a·b^t with a > 0? Select all that apply.

Explanation

Growth requires b > 1 and a > 0. A and E satisfy this. B has a = 0 (degenerate). C has b

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20) To solve y = a·b^t for t (with a > 0 and b > 0, b ≠ 1), t = ____.

Explanation

Divide by a to get y/a = b^t. Taking natural logs: ln(y/a) = t·ln b ⇒ t = ln(y/a)/ln b.

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In N(t) = a·b^t, the coefficient a equals the value at t = 0.
A culture has P(0) = 200 and P(1) = 260. Assuming P(t) = a·b^t, which...
For b = 1.2 in N(t) = a·b^t, the exact doubling time (in periods) is:
If b = 1 in N(t) = a·b^t, then N(t) stays constant over time.
Which model matches “starts at 1000 and grows by 5% each hour”?
In N(t) = a·b^{t/3}, the multiplier over 3 time units is b.
If b = 0.8 in y = a·b^t, what is y after two periods relative to a?
For N(t) = 300(1.25)^t, what does the coefficient 300 represent?
In N(t) = 300(1.25)^t, what does 1.25 represent?
For N(t) = a·b^t with b = 1.25, what is the percent rate of change...
Suppose P(t) = a·b^t and P(3) = 2a. What is b?
For N(t) = 120(1.5)^t, compute N(2).
For P(t) = 1200(0.9)^t, the percent change per period is ____.
Which statements about M(t) = 500(1.03)^t are true? Select all that...
For S(t) = 800(0.75)^t, S(0) equals ____.
For y(t) = a·b^t with a > 0 and 0 < b < 1, which are true?...
Given data follow an exponential model with P(0) = 50 and P(1) = 60....
In N(t) = a·b^t, the percent change per period equals ____%.
Which pairs (a,b) represent exponential growth for y = a·b^t with a...
To solve y = a·b^t for t (with a > 0 and b > 0, b ≠ 1), t =...
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