Exponential Functions & Growth — Practice
Linear growth adds the same amount each step. Exponential growth multiplies by the same amount — and always wins in the end.
What these questions 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.
Problem 1
Is 4, 12, 36, 108 linear or exponential?
Worked answer
exponential
12 ÷ 4 = 3
36 ÷ 12 = 3
Constant ratio, so exponential
Problem 2
Is 4, 11, 18, 25 linear or exponential?
Worked answer
linear
11 - 4 = 7
18 - 11 = 7
Constant difference, so linear
Problem 3
For f(x) = 3 · 2x, find f(4).
Worked answer
48
24 = 16
3 × 16 = 48
Problem 4
For f(x) = 7 · 5x, find f(0).
Worked answer
7
50 = 1
7 × 1 = 7
Problem 5
Growing by 6% means multiplying by what factor?
Worked answer
1.06
1 + 0.06 = 1.06
Problem 6
Shrinking by 12% means multiplying by what factor?
Worked answer
0.88
1 - 0.12 = 0.88
Problem 7
A population of 4000 grows 5% a year. Write P(t).
Worked answer
P(t) = 4000(1.05)^t
P(t) = 4000(1.05)t
Problem 8
A car worth $18,000 loses 20% a year. Write V(t).
Worked answer
V(t) = 18000(0.8)^t
V(t) = 18000(0.8)t
Problem 9
Write the exponential rule for the table x: 0, 1, 2, 3 and y: 6, 18, 54, 162.
Worked answer
y = 6 * 3^x
a = 6
Ratio = 3
y = 6 · 3x
Problem 10
A $500 investment grows 8% a year. What is it worth after 3 years, to the nearest cent?
Worked answer
629.86
1.083 = 1.259712
500 × 1.259712 = 629.856
≈ $629.86
Problem 11
As x grows without bound, the graph of y = 3(0.5)x
- approaches 0 but never reaches it
- increases without bound
- becomes negative
- reaches 0 exactly
Worked answer
approaches 0 but never reaches it
Each step halves the value
It gets arbitrarily small
The x-axis is an asymptote
Problem 12
Find the 6th term of the geometric sequence 2, 6, 18, ….
Worked answer
486
2 · 35
= 2 × 243 = 486
Problem 13
A bacteria count doubles every hour, starting at 50. How many after 8 hours?
Worked answer
12800
28 = 256
50 × 256 = 12800
Problem 14
For how many years must $1000 grow at 10% to first exceed $2000? Give a whole number of years.
Worked answer
8
(1.1)7 = 1.949, so $1948.72 — not yet
(1.1)8 = 2.144, so $2143.59
8 years
Problem 15
Function A is y = 100x and B is y = 2x. At x = 10, which is larger?
Worked answer
B
A: 100(10) = 1000
B: 210 = 1024
1024 > 1000, so B — the exponential has just overtaken
Problem 16
A drug leaves the body at 25% per hour. Starting from 80 mg, how much remains after 3 hours, to the nearest tenth?
Worked answer
33.8
80(0.75)3
0.753 = 0.421875
80 × 0.421875 = 33.75 ≈ 33.8 mg