An exponential function has the form y = a(b)ˣ, where a is the starting value and b is the growth or decay factor. When b is greater than 1, the function models growth — values increase, getting faster over time. When b is between 0 and 1, the function models decay — values decrease, slowing down over time as they approach (but never quite reach) zero.
Example
A population of bacteria starting at 100 and doubling every hour is modelled by P(t) = 100(2)ᵗ, where t is hours elapsed — after 3 hours, P(3) = 100(2)³ = 800 bacteria.
Key terms
Exponential growth:
Repeated multiplication by a factor greater than 1, causing accelerating increase.
Exponential decay:
Repeated multiplication by a factor between 0 and 1, causing slowing decrease.
Growth/decay factor:
The base (b) a starting value is repeatedly multiplied by.
Questions
1. An exponential function has the form:
y = a(b)ˣ
y = ax + b
y = ax²
y = a + x
2. When b is greater than 1, the function models:
Growth
Decay only
No change at all
A straight line
3. When b is between 0 and 1, the function models:
Decay
Growth only
No change at all
A straight line
4. In y = a(b)ˣ, "a" represents:
The starting value
The growth factor only
The final answer, always
A type of punctuation
5. If P(t) = 100(2)ᵗ, the starting value is:
100
2
0
t
6. Exponential decay values approach:
Zero, but never quite reach it
Infinity, immediately
A negative number always
Nothing in particular
7. Exponential growth causes values to increase:
Getting faster over time
At a constant, unchanging rate
Getting slower over time
Only once, then stopping
8. If P(t) = 100(2)ᵗ, what is P(3)?
800
600
300
200
9. If a population starts at 50 and doubles every year, modelled as P(t) = 50(2)ᵗ, what is P(2)?
200
100
150
250
10. A car worth $20,000 depreciating at 10% per year could be modelled with a decay factor of:
0.9
1.1
0.1
10
11. If V(t) = 20000(0.9)ᵗ, what is V(1), rounded to the nearest dollar?
$18,000
$20,000
$2,000
$22,000
12. Why does exponential growth eventually outpace linear growth, even if linear growth starts faster?
Exponential growth compounds on itself, so its rate of increase keeps accelerating
Linear growth always eventually overtakes exponential growth
Exponential and linear growth are mathematically identical over time
Exponential growth always starts and stays the fastest from the very beginning
13. A decay factor closer to 0 (e.g. 0.5 vs 0.9) causes values to decrease:
More quickly
More slowly
At exactly the same rate regardless
Not at all
14. If bacteria triple every hour starting from 10, modelled as P(t) = 10(3)ᵗ, what is P(2)?
90
60
30
100
15. Why might exponential decay never reach exactly zero, even after a very long time?
Each step multiplies the remaining value by a fraction, which shrinks it but never eliminates it completely
Decay always reaches exactly zero after a fixed, predictable number of steps
Exponential decay is mathematically identical to subtraction
Multiplying by a fraction always eventually produces a negative number
16. A savings account earning compound interest and a population doubling each generation are both examples of:
Exponential growth
Exponential decay
Linear growth only
No mathematical pattern at all
17. If a substance decays with V(t) = 500(0.8)ᵗ, approximately what is V(3)?
256
400
320
160
18. Why is understanding exponential functions important for interpreting real-world claims about growth (e.g. "cases doubled every week")?
Exponential trends can escalate dramatically and unintuitively compared to linear expectations
Doubling claims always describe linear, not exponential, growth
Exponential trends always slow down immediately after being reported
Real-world growth claims never follow exponential patterns
19. Why might comparing two investment options — one with linear growth and one with exponential growth — favour the exponential option over a long time period?
Compounding growth accelerates over time, eventually producing a much larger total than steady linear increases
Linear growth always produces a larger total over any period of time
Both growth types always produce identical totals eventually
Exponential growth only applies to very short time periods
20. If a car valued at $30,000 depreciates by 15% each year, modelled as V(t) = 30000(0.85)ᵗ, approximately what is V(2)?
$21,675
$25,500
$27,000
$19,500
21. Why might a graph of exponential growth appear to rise slowly at first before rising very steeply later?
Each multiplication builds on an already-larger value, so the rate of increase itself keeps growing
Exponential graphs always rise at a perfectly constant rate throughout
Exponential growth is identical to a straight line on a graph
The steepness of an exponential graph never changes over time
Answer key (parent copy)
1. y = a(b)ˣ
2. Growth
3. Decay
4. The starting value
5. 100
6. Zero, but never quite reach it
7. Getting faster over time
8. 800
9. 200
10. 0.9
11. $18,000
12. Exponential growth compounds on itself, so its rate of increase keeps accelerating
13. More quickly
14. 90
15. Each step multiplies the remaining value by a fraction, which shrinks it but never eliminates it completely
16. Exponential growth
17. 256
18. Exponential trends can escalate dramatically and unintuitively compared to linear expectations
19. Compounding growth accelerates over time, eventually producing a much larger total than steady linear increases
20. $21,675
21. Each multiplication builds on an already-larger value, so the rate of increase itself keeps growing