Exponential Functions & Growth — Drills
Exponential Functions & Growth has 12 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills cover
These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Ratio — A comparison of two quantities, like 3 cups flour to 2 cups water.
- Base — The number being multiplied. In 53 the base is 5.
- Factor — A number that divides evenly into another.
- Function — A rule pairing each input with exactly one output.
- Asymptote — A line the graph approaches but never reaches — here, the x-axis.
- Sequence — An ordered list of numbers. The nth one is an.
Classifying
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Is 5, 14, 23, 32 linear or exponential?
Worked answer
linear
Differences: 9, 9, 9
Ratios: 2.8, 1.6428571428571428, 1.391304347826087
Constant difference 9, so linear.
Problem 2
Is 4, 10, 16, 22 linear or exponential?
Worked answer
linear
Differences: 6, 6, 6
Ratios: 2.5, 1.6, 1.375
Constant difference 6, so linear.
Evaluating
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
For f(x) = 6 · 4x, find f(5).
Worked answer
6144
45 = 1024
6 × 1024 = 6144
Problem 2
For f(x) = 8 · 3x, find f(3).
Worked answer
216
33 = 27
8 × 27 = 216
Initial
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
For f(x) = 35 · 5x, find f(0).
Worked answer
35
50 = 1
35 × 1 = 35
Problem 2
For f(x) = 5 · 4x, find f(0).
Worked answer
5
40 = 1
5 × 1 = 5
Factoring
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Shrinking by 8% means multiplying by what factor?
Worked answer
0.92
1 - 0.08 = 0.92
Problem 2
Growing by 18% means multiplying by what factor?
Worked answer
1.18
1 + 0.18 = 1.18
Model
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A quantity starts at 4900 and decays 21% a year. Write P(t).
Worked answer
P(t) = 4900(0.79)^t
a = 4900
b = 0.79
P(t) = 4900(0.79)t
Problem 2
A quantity starts at 7100 and grows 17% a year. Write P(t).
Worked answer
P(t) = 7100(1.17)^t
a = 7100
b = 1.17
P(t) = 7100(1.17)t
Rule
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Write the exponential rule for x: 0, 1, 2, 3 and y: 9, 36, 144, 576.
Worked answer
y = 9 * 4^x
a = 9
Ratio = 4
y = 9 · 4x
Problem 2
Write the exponential rule for x: 0, 1, 2, 3 and y: 2, 8, 32, 128.
Worked answer
y = 2 * 4^x
a = 2
Ratio = 4
y = 2 · 4x
Compute
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A $1800 investment grows 6% a year. What is it worth after 3 years, to the nearest cent?
Worked answer
2143.83
(1.06)3 = 1.191016
1800 × 1.191016 ≈ 2143.83
Problem 2
A $300 investment grows 5% a year. What is it worth after 5 years, to the nearest cent?
Worked answer
382.88
(1.05)5 = 1.276282
300 × 1.276282 ≈ 382.88
From a graph
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
As x grows without bound, the graph of y = 6(0.5)x
- increases without bound
- becomes negative
- approaches 0 but never reaches it
- reaches 0 exactly
Worked answer
approaches 0 but never reaches it
0.5 < 1, so this is decay.
Each step multiplies by less than one.
The x-axis is an asymptote.
Problem 2
As x grows without bound, the graph of y = 8(0.5)x
- approaches 0 but never reaches it
- reaches 0 exactly
- becomes negative
- increases without bound
Worked answer
approaches 0 but never reaches it
0.5 < 1, so this is decay.
Each step multiplies by less than one.
The x-axis is an asymptote.
Geometric
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the 7th term of the geometric sequence 3, 12, 48, ….
Worked answer
12288
3 · 46
46 = 4096
= 12288
Problem 2
Find the 4th term of the geometric sequence 5, 15, 45, ….
Worked answer
135
5 · 33
33 = 27
= 135
Double
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A count triples every hour, starting at 17. How many after 8 hours?
Worked answer
111537
38 = 6561
17 × 6561 = 111537
Problem 2
A count doubles every hour, starting at 36. How many after 7 hours?
Worked answer
4608
27 = 128
36 × 128 = 4608
Solving
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
For how many years must $6000 grow at 12% before it first exceeds $12000? Give a whole number of years.
Worked answer
7
At t = 6: $11842.94 — not yet
At t = 7: $13264.09
7 years
Problem 2
For how many years must $8000 grow at 10% before it first exceeds $16000? Give a whole number of years.
Worked answer
8
At t = 7: $15589.74 — not yet
At t = 8: $17148.71
8 years
Comparing
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Function A is y = 200x and B is y = 2x. At x = 10, which is larger?
- A
- Cannot be determined
- They are equal
- B
Worked answer
A
A: 200(10) = 2000
B: 210 = 1024
2000 > 1024, so A.
Problem 2
Function A is y = 50x and B is y = 2x. At x = 10, which is larger?
- B
- A
- They are equal
- Cannot be determined
Worked answer
B
A: 50(10) = 500
B: 210 = 1024
1024 > 500, so B.