Exponential Functions & Growth — Print

Linear growth adds the same amount each step. Exponential growth multiplies by the same amount — and always wins in the end.

What the practice 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.

A king offers a reward: one grain of rice on the first square of a chessboard, doubling on each square after.

It sounds modest. One, two, four, eight — you could carry the first dozen squares in a cup.

one grain on the first square, then double every time 1248163264128 square 8 already needs 128 grains, and the board has 64 squares

By square 8 you are at 128 grains. Adding one grain each time would have reached 8.

By square 20 it is half a million. By square 64 the pile is larger than a mountain — more rice than has ever been grown.

And nothing dramatic happened at any single step. Every step was just "times two."

That is exponential growth: it looks slow at the start and then suddenly does not. Linear growth adds a constant; exponential growth multiplies by one — and multiplying always wins in the end.

Linear growth against exponential growth adding the same amount, against multiplying by the same amount + 6 each step x 2 each step time amount linear: y = mx + b exponential: y = a(b)^x the exponential starts behind and still overtakes, every single time

Spotting exponential behaviour

In a table with evenly spaced x values: if consecutive y values have a constant difference, it is linear. If they have a constant ratio, it is exponential.

3, 6, 12, 24 has ratio 2 every time, so it is exponential with b = 2.

3, 6, 9, 12 has difference 3 every time, so it is linear.

Differences or ratios

Subtract to test for linear. Divide to test for exponential. Do both on any unfamiliar table and one of them will come out constant.

Percentages and the growth factor

Growing by 8% means multiplying by 1.08, because you keep the original 100% and add 8% more.

Shrinking by 8% means multiplying by 0.92.

So b = 1 + r for growth and b = 1 - r for decay, with r written as a decimal.

Do not use 0.08 as the base

Multiplying by 0.08 destroys 92% of the amount each step. Growth of 8% needs the factor 1.08.

The graph

Growth curves rise slowly at first and then steeply — always increasing, always concave up.

Decay curves fall fast and then level off, approaching the x-axis without ever touching it.

The y-intercept is always a, since b0 = 1 whatever b is.

Check x = 0 first

Whatever the base, b0 = 1, so f(0) = a. That gives you one point of the graph with no work at all.

Linear against exponential

Exponential growth eventually overtakes any linear growth, however gentle the base or steep the line.

Over small ranges a linear model may fit exponential data acceptably, which is why short-term data can mislead.

Doubling every step from 1 reaches a million in twenty steps. Adding one reaches twenty.

Geometric sequences

A geometric sequence has a constant ratio: an = a1 · rn-1.

It is an exponential function restricted to the whole numbers.

Recursively: an = r · an-1, starting from a1.

Worked examples

Table or not

Is 5, 15, 45, 135 linear or exponential? Write the rule.

  1. Differences: 10, 30, 90 — not constant. So not linear.
  2. Ratios: 155 = 3, 4515 = 3, 13545 = 3 Constant ratio, so exponential.
  3. At x = 0 the value would be 5, so a = 5 and b = 3. Treating 5 as the initial term.
  4. f(x) = 5 · 3x Check f(2) = 5(9) = 45 ✓

f(x) = 5 · 3x

Percentage growth

A town of 12,000 grows 3% a year. Write a model and find the population after 10 years.

  1. a = 12000 and b = 1 + 0.03 = 1.03 Keep the original, add 3%.
  2. P(t) = 12000(1.03)t With t in years.
  3. P(10) = 12000(1.03)10 ≈ 12000(1.3439) Raise to the tenth power.
  4. ≈ 16,127 people Sensible: about a third more after a decade.

P(t) = 12000(1.03)t; about 16,127

Decay

A car worth $24,000 loses 15% of its value each year. What is it worth after 4 years?

  1. Losing 15% leaves 85%, so b = 0.85. 1 - 0.15 = 0.85.
  2. V(t) = 24000(0.85)t Exponential decay.
  3. V(4) = 24000(0.85)4 = 24000(0.52200625) Four years of shrinking.
  4. ≈ $12,528 A little over half, which matches losing 15% four times.

About $12,528

Words to know

Exponential function
f(x) = a · bx with a ≠ 0, b > 0, b ≠ 1.
Growth factor
b itself. b > 1 means growth, 0 < b < 1 means decay.
Initial value
a, the output when x = 0.
Growth rate
The percentage change. b = 1 + r for growth, b = 1 - r for decay.
Asymptote
A line the graph approaches but never reaches — here, the x-axis.

Mistakes to watch for

Using the percentage as the base. Growth of 8% means multiplying by 1.08, because you keep the original 100% and add 8 more. Multiplying by 0.08 destroys 92% of the amount every step.

Testing only for a constant difference. If the differences are not constant, try the ratios — a constant ratio means exponential. Checking only one of the two makes exponential data look like an error.

Multiplying the base by the exponent. 3 · 24 is 3 × 16 = 48, not 64. The exponent attaches to the 2 alone, and the 3 is a separate multiplier waiting its turn.

Believing a decay curve reaches zero. Halving again and again gets arbitrarily small but never arrives. That is what an asymptote is, and it is why a drug never completely leaves the body.

The rice-and-chessboard story appears in Indian and Persian sources from at least the 11th century, usually ending with the king realising he cannot pay.

The total is 264 - 1 grains: about 18 quintillion, roughly a thousand years of world rice production.

The same arithmetic explains compound interest, viral spread and the growth of computing power — one idea, wearing very different clothes.

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

  1. approaches 0 but never reaches it
  2. increases without bound
  3. becomes negative
  4. 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

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