Exponential Functions & Growth — Challenge
Linear growth adds the same amount each step. Exponential growth multiplies by the same amount — and always wins in the end.
What these problems 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
If y=2· 3x, what is y when x=2?
- 12
- 18
- 36
- 6
- 9
Worked answer
18
Work out the power before multiplying: 32 = 9.
Then y = 2· 9 = 18.
Multiplying the base by 2 first would give 62 = 36, which is a different rule.
Problem 2
A colony of 500 bacteria doubles every year. How many are there after 3 years?
- 1500
- 3000
- 4000
- 8000
- 2000
Worked answer
4000
Doubling three times multiplies the population by 23 = 8.
500· 8 = 4000 bacteria.
Multiplying by 3 and 2 to get 3000 treats the growth as repeated addition rather than repeated doubling.
Problem 3
If y=100(0.5)x, what is y when x=3?
- 12.5
- 25
- 50
- 150
- 6.25
Worked answer
12.5
(0.5)3 = 0.125, which is halving three times.
100· 0.125 = 12.5.
Halving once gives 50 and twice gives 25, so the third halving is needed.
Problem 4
Which of these describes exponential growth?
- y=3x
- y=3x
- y=x3
- y=3
- y=x+3
Worked answer
y=3x
Exponential growth has the variable in the exponent, so the quantity multiplies by a fixed factor each step.
In y = 3x the x is the exponent, so y triples whenever x rises by 1.
In y = x3 the variable is the base instead, which grows steadily rather than exponentially, and y = 3x is linear.
Problem 5
What is 50?
- 0
- 1
- 5
- 25
- Undefined
Worked answer
1
Dividing 51 by 5 gives 50, and 55 = 1.
So any non-zero number to the power 0 is 1.
This is why an exponential graph always passes through (0,a) for y = a· bx.
Problem 6
An investment of $1000 grows by 10% each year. What is it worth in dollars after 2 years?
- $1100
- $1200
- $1210
- $1220
- $1300
Worked answer
$1210
Each year multiplies the value by 1.10, so two years multiply by 1.102 = 1.21.
1000· 1.21 = 1210 dollars.
Adding 10% of the original twice would give 1200, which misses the interest earned on the first year's interest.
Problem 7
In the equation y=a· bx, what is b called?
- The base
- The exponent
- The intercept
- The slope
- The coefficient
Worked answer
The base
The quantity being raised to a power is the base, and here that is b.
The x is the exponent, and a is the starting value, found by putting x=0.
The base decides the factor by which y changes at each step.
Problem 8
A substance halves every 5 years. What fraction of it remains after 15 years?
- 12
- 13
- 18
- 115
- 35
Worked answer
18
Fifteen years contains 155 = 3 halving periods.
Halving three times leaves (12)3 = 18.
Dividing by 3 to get 13 would treat the decay as linear rather than as repeated halving.