Exponential Functions

An exponential function describes a relationship in which a quantity changes by the same factor or percentage over equal intervals.

A common form is:

y = abˣ

where:

• a = initial value
• b = growth or decay factor
• x = independent variable
• y = dependent variable

The important feature is that the variable x appears in the exponent.

For example:

y = 3(2ˣ)

is exponential because x is an exponent.

By contrast:

y = 3x + 2

is linear because x is multiplied by a constant.

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Recognizing an Exponential Function

Consider:

y = 5(2ˣ)

The variable x appears in the exponent.

Therefore this is an exponential function.

Now consider:

y = 5x²

Here x is the base and 2 is the exponent.

This is quadratic, not exponential.

The position of the variable matters.

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Exponential Growth

An exponential function represents growth when its values repeatedly multiply by a factor greater than 1.

For:

y = abˣ

exponential growth occurs when:

b > 1

For example:

y = 100(2ˣ)

has:

a = 100

b = 2

The quantity doubles every time x increases by 1.

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A Growth Table

Consider:

y = 2ˣ

Notice:

1 → 2 → 4 → 8 → 16 → 32

Each value is multiplied by:

2

This constant multiplication factor is the key sign of an exponential relationship.

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Constant Factor Rather Than Constant Difference

For exponential functions, look for a constant ratio between consecutive outputs.

In:

3, 6, 12, 24, 48

each value is:

2 times the previous value

Therefore:

growth factor = 2

Compare this with:

3, 6, 9, 12, 15

Here we repeatedly add 3.

That is a linear pattern.

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Exponential Decay

An exponential function represents decay when the quantity repeatedly decreases by the same percentage or multiplication factor.

For:

y = abˣ

decay occurs when:

0 < b < 1

For example:

y = 100(0.5ˣ)

means the quantity is multiplied by:

0.5

each interval.

In other words, it halves each time.

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A Decay Table

Consider:

y = 80(0.5ˣ)

The values are:

80 → 40 → 20 → 10 → 5

Each value is half the previous value.

The decay factor is:

0.5

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Initial Value

In:

y = abˣ

the value a represents the initial value when:

x = 0

This works because:

b⁰ = 1

Therefore:

y = a(1)

y = a

For:

y = 250(1.4ˣ)

the initial value is:

250

The graph passes through:

(0, 250)

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Growth Factor

In:

y = abˣ

the value b is the growth factor or decay factor.

For:

y = 50(3ˣ)

the growth factor is:

3

The quantity triples every interval.

For:

y = 50(0.8ˣ)

the decay factor is:

0.8

The quantity retains 80% of its previous value each interval.

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Growth Rate as a Percentage

Suppose a quantity increases by:

20%

each year.

The original amount represents:

100%

After the increase, it becomes:

120%

As a decimal:

120% = 1.20

Therefore the growth factor is:

1.20

The model has the form:

y = a(1.20ˣ)

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Growth Factor Formula

If the growth rate is r as a decimal:

growth factor = 1 + r

For example:

10% growth:

b = 1 + 0.10 = 1.10

25% growth:

b = 1 + 0.25 = 1.25

4% growth:

b = 1 + 0.04 = 1.04

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Decay Factor from a Percentage

Suppose a quantity decreases by:

20%

each year.

It retains:

80%

of its previous value.

Therefore:

b = 0.80

The model has the form:

y = a(0.80ˣ)

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Decay Factor Formula

If the decay rate is r:

decay factor = 1 − r

For example:

10% decrease:

b = 1 − 0.10 = 0.90

25% decrease:

b = 1 − 0.25 = 0.75

60% decrease:

b = 1 − 0.60 = 0.40

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Growth Rate Is Not the Same as Growth Factor

This distinction is important.

If something increases by:

15%

then:

growth rate = 15% = 0.15

but:

growth factor = 1.15

For:

y = 500(1.15ˣ)

the number:

1.15

does not mean 115% growth.

It means the quantity retains its original 100% and gains an additional 15%.

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Decay Rate Is Not the Same as Decay Factor

For:

y = 500(0.85ˣ)

the decay factor is:

0.85

This means:

85% remains

so:

15% is lost

each interval.

Therefore:

decay rate = 15%

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Shape of an Exponential Growth Graph

An exponential growth graph typically:

• increases slowly at first
• becomes increasingly steep
• rises rapidly for larger x-values
• curves upward

Unlike a linear graph, it does not increase by equal amounts over equal intervals.

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Shape of an Exponential Decay Graph

An exponential decay graph typically:

• decreases quickly at first
• decreases more slowly later
• approaches zero
• does not normally reach zero in the basic mathematical model

This produces a curved graph that becomes flatter over time.

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Horizontal Asymptote

For basic exponential functions such as:

y = 2ˣ

and:

y = 0.5ˣ

the graph approaches:

y = 0

but does not reach it.

The line:

y = 0

is called a horizontal asymptote.

An asymptote is a line that a graph approaches increasingly closely.

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Exponential Functions Are Not Straight Lines

Consider:

y = 2ˣ

The slope is constantly changing.

As x increases, the graph becomes steeper.

Therefore the graph is curved.

This differs fundamentally from a linear function, which has a constant slope.

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Linear Growth

A linear function has the form:

y = mx + b

Linear growth involves adding the same amount during each interval.

For example:

100, 120, 140, 160, 180

Each value increases by:

20

The difference is constant.

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Exponential Growth

Exponential growth involves multiplying by the same factor.

For example:

100, 120, 144, 172.8, 207.36

Each value is multiplied by:

1.2

This represents:

20% growth

during each interval.

The amount being added gets larger because 20% is being calculated from an increasingly large quantity.

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Linear Versus Exponential Growth

Suppose two quantities both begin at 100.

Linear:

L = 100 + 20x

Exponential:

E = 100(1.2ˣ)

At first, the values may appear similar.

Over time, the exponential function grows much faster.

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Why Exponential Growth Becomes So Fast

Suppose a population increases by:

10% each year

The increase itself becomes larger every year.

If the population is:

1,000

the first increase is:

100

giving:

1,100

The next 10% increase is based on 1,100:

110

giving:

1,210

The next increase is:

121

The quantity being added is itself increasing.

This is the key reason exponential growth can become extremely rapid.

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Repeated Addition Versus Repeated Multiplication

This provides a useful recognition rule.

Linear

Repeated addition:

5, 8, 11, 14, 17

Add:

3

each time.

Exponential

Repeated multiplication:

5, 10, 20, 40, 80

Multiply by:

2

each time.

Think:

Linear → constant difference

Exponential → constant ratio

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Recognizing Exponential Functions from Tables

Consider:

Calculate consecutive ratios:

12/4 = 3

36/12 = 3

108/36 = 3

324/108 = 3

The ratio is constant.

Therefore the relationship is exponential.

The model is:

y = 4(3ˣ)

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Recognizing Exponential Decay from Tables

Consider:

Ratios:

80/160 = 0.5

40/80 = 0.5

20/40 = 0.5

10/20 = 0.5

The constant ratio is:

0.5

Therefore this is exponential decay.

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Linear, Quadratic, and Exponential Tables

These patterns can often be distinguished by their differences or ratios.

Linear

Constant first difference

Example:

2, 5, 8, 11, 14

Quadratic

Constant second difference

Example:

0, 1, 4, 9, 16

Exponential

Constant ratio

Example:

2, 6, 18, 54, 162

This is an extremely useful recognition strategy.

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A Comparison Table

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Population Growth

Populations can sometimes grow approximately exponentially when the percentage growth rate remains relatively constant.

Suppose a bacterial population begins with:

500 bacteria

and increases by:

20% each hour

The growth factor is:

1.20

Model:

P = 500(1.2ᵗ)

where t is time in hours.

After 4 hours:

P = 500(1.2⁴)

P = 500(2.0736)

P = 1036.8

The model predicts approximately:

1,037 bacteria

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Doubling

A common exponential growth pattern is doubling.

Suppose a cell population doubles every hour.

Starting with:

200 cells

the pattern is:

200 → 400 → 800 → 1600 → 3200

The model is:

P = 200(2ᵗ)

The factor 2 represents doubling.

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Compound Interest

Money invested with compound interest can grow exponentially.

Suppose $1,000 grows by:

5% per year

The growth factor is:

1.05

Model:

A = 1000(1.05ᵗ)

After 10 years:

A = 1000(1.05¹⁰)

Approximately:

A = $1,628.89

The interest earned in earlier years also earns interest in later years.

This is why compound interest is exponential rather than linear.

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Simple Versus Compound Growth

If $100 is added to an account every year:

linear growth

because the same amount is added.

If the account increases by 5% every year:

exponential growth

because the same percentage is applied repeatedly.

The wording of a problem often reveals the type of relationship.

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Inflation

If prices increase by approximately the same percentage each year, an exponential model may be useful.

For example, if a price begins at $200 and increases by 3% annually:

P = 200(1.03ᵗ)

This does not mean exactly $6 is added every year.

The amount added increases as the price becomes larger.

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Radioactive Decay

Radioactive materials decrease exponentially.

Suppose a sample contains:

800 mg

and half remains after each half-life.

The pattern is:

800 → 400 → 200 → 100 → 50

A simple model using n as the number of half-lives is:

M = 800(0.5ⁿ)

This is exponential decay.

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Half-Life

Half-life is the time required for half of a quantity to decay.

After:

1 half-life → 1/2 remains

2 half-lives → 1/4 remains

3 half-lives → 1/8 remains

4 half-lives → 1/16 remains

The same fraction disappears repeatedly, producing exponential decay.

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Depreciation

The value of some products decreases by a percentage each year.

Suppose a car is worth:

$30,000

and loses:

15% of its value each year

It retains:

85%

Therefore:

V = 30000(0.85ᵗ)

After one year:

V = $25,500

After another year, the 15% decrease is calculated from $25,500 rather than the original $30,000.

This produces exponential decay.

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Medication in the Body

Some medications are removed from the body at approximately exponential rates.

If 80% remains after each hour, then:

decay factor = 0.80

Starting with 200 mg:

A = 200(0.8ᵗ)

The amount decreases rapidly at first and more slowly later.

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Cooling and Exponential Relationships

Under suitable conditions, the difference between an object's temperature and its surroundings can decrease approximately exponentially.

This means cooling does not necessarily occur by exactly the same number of degrees each minute.

The temperature often changes rapidly at first and more slowly as it approaches the surrounding temperature.

This is another example of exponential decay toward a limiting value.

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Interpreting Exponential Graphs

When examining an exponential graph, look for:

• initial value
• growth or decay
• rate or factor
• direction of change
• steepness
• asymptotic behaviour
• meaningful domain
• real-world interpretation

The shape of the graph provides important information about the relationship.

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Reading the Initial Value

For:

y = 300(1.4ˣ)

when:

x = 0

y = 300

Therefore the graph crosses the y-axis at:

(0, 300)

This represents the initial quantity.

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Reading Growth from a Graph

If an exponential graph increases:

• slowly at first
• then increasingly rapidly

it may represent exponential growth.

If equal horizontal intervals correspond to multiplication by approximately the same factor, this provides stronger evidence.

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Reading Decay from a Graph

If a graph:

• decreases rapidly initially
• becomes increasingly flat
• approaches a horizontal value

it may represent exponential decay.

The graph's behaviour reflects repeated percentage or proportional change.

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Exponential Models and Asymptotes

Consider:

y = 100(0.5ˣ)

As x increases:

y gets closer and closer to 0

but the basic mathematical function does not reach zero for any finite x.

This is different from a linear decay model such as:

y = 100 − 10x

which reaches zero exactly when:

x = 10

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Comparing Linear and Exponential Decay

Linear decay:

100, 90, 80, 70, 60

Subtract:

10

each time.

Exponential decay:

100, 90, 81, 72.9, 65.61

Multiply by:

0.9

each time.

Linear decay removes the same amount.

Exponential decay removes the same percentage.

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Predicting Values

Suppose:

P = 200(1.5ᵗ)

Predict P when:

t = 4

Substitute:

P = 200(1.5⁴)

P = 200(5.0625)

P = 1012.5

The model predicts approximately:

1,013

if the quantity must be a whole number.

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Working Backwards Conceptually

Suppose:

P = 100(2ᵗ)

The values are:

t = 0 → 100

t = 1 → 200

t = 2 → 400

t = 3 → 800

If:

P = 800

we can recognize:

t = 3

More advanced exponential equations may require logarithms to solve when the answer cannot be identified easily.

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Comparing Two Exponential Functions

Consider:

A = 100(1.1ˣ)

B = 100(1.3ˣ)

Both begin at:

100

But B has the larger growth factor.

Therefore B grows faster for positive x.

The difference between them becomes increasingly large over time.

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Same Growth Factor, Different Initial Values

Consider:

A = 100(1.2ˣ)

B = 500(1.2ˣ)

Both grow by:

20% per interval

but B begins five times larger.

In fact, B remains five times A for every x-value.

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Comparing Growth Rates Carefully

Consider:

A = 100(1.10ˣ)

B = 20(1.30ˣ)

A begins much larger.

B grows faster.

Therefore A may initially have the greater value, while B may eventually overtake it.

When comparing exponential functions, consider both:

• initial value
• growth factor

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Exponential Models Have Limits

A mathematical exponential function may continue indefinitely.

Real systems usually cannot.

A bacterial population cannot grow exponentially forever because eventually there may be shortages of:

• food
• space
• oxygen
• other resources

Therefore exponential growth may be a good model only over a limited interval.

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Population Growth Is Not Always Exponential

Real populations may initially grow approximately exponentially.

Later, growth may slow because of:

• limited resources
• disease
• competition
• environmental changes
• migration

A different model may then become more appropriate.

Mathematical models must always be evaluated against reality.

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Exponential Relationships in Science

Exponential functions appear in many scientific contexts, including:

• radioactive decay
• population growth
• bacterial growth
• charging and discharging processes
• attenuation
• some cooling processes
• drug elimination
• chain reactions

Recognizing exponential behaviour helps scientists make predictions about rapidly changing systems.

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Exponential Relationships in Finance

Exponential functions are also important in:

• compound interest
• inflation
• investment growth
• depreciation
• debt growth

Small percentage changes repeated many times can produce large long-term effects.

This is one reason exponential growth is so important in financial decision-making.

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Recognizing Exponential Relationships from Words

Words suggesting exponential growth include:

• doubles every...
• triples every...
• increases by 5% each...
• grows by 20% per...
• multiplied by...

Words suggesting exponential decay include:

• halves every...
• loses 10% each...
• decreases by 25% per...
• retains 80% each...
• decays by...

The key idea is repeated multiplication, not repeated addition or subtraction.

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A Reliable Recognition Strategy

Use:

Equation → Table → Graph → Context

Equation

Is the variable in the exponent?

Table

Is there approximately a constant ratio between outputs for equally spaced inputs?

Graph

Does it show the characteristic exponential growth or decay curve?

Context

Does the situation involve repeated percentage or proportional change?

Using several types of evidence gives a stronger conclusion.

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Linear, Quadratic, or Exponential?

Ask:

Is there a constant first difference?

Likely:

Linear

Is there a constant second difference?

Likely:

Quadratic

Is there a constant ratio?

Likely:

Exponential

This applies when the x-values are equally spaced.

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Common Mistakes

Mistake: Thinking y = x² is exponential.

The variable is the base, not the exponent.

y = x² is quadratic.

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Mistake: Thinking y = 2ˣ is quadratic.

The variable appears in the exponent.

Therefore it is exponential.

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Mistake: Looking for a constant difference in an exponential table.

Look for a constant ratio.

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Mistake: Confusing growth rate and growth factor.

20% growth gives:

growth rate = 0.20

growth factor = 1.20

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Mistake: Using 0.20 as the factor for 20% growth.

That would mean only 20% remains each interval.

For 20% growth, use:

1.20

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Mistake: Using 1.20 for 20% decay.

For 20% decay:

80% remains

so use:

0.80

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Mistake: Thinking exponential growth adds the same amount each time.

It multiplies by the same factor.

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Mistake: Thinking exponential decay subtracts the same amount each time.

It usually decreases by the same percentage or factor.

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Mistake: Assuming exponential decay reaches zero quickly.

Basic exponential decay approaches zero asymptotically.

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Mistake: Assuming exponential growth can continue forever in real systems.

Resources and physical limitations often restrict exponential growth.

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Mistake: Comparing exponential models using only their initial values.

The growth factors must also be considered.

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Mistake: Assuming every rapidly increasing graph is exponential.

Check the equation, ratios, and context before classifying the relationship.

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Key Terms

Exponential function — function in which the independent variable appears in an exponent.

Exponential growth — repeated proportional increase represented by a growth factor greater than 1.

Exponential decay — repeated proportional decrease represented by a factor between 0 and 1.

Initial value — starting amount, represented by a in y = abˣ.

Growth factor — multiplier greater than 1 applied during each interval.

Decay factor — multiplier between 0 and 1 applied during each interval.

Growth rate — percentage by which a quantity increases during each interval.

Decay rate — percentage by which a quantity decreases during each interval.

Constant ratio — same quotient between consecutive outputs for equally spaced inputs.

Linear growth — growth involving a constant additive change.

Compound growth — growth in which increases are repeatedly applied to the new total.

Compound interest — interest calculated on both the original amount and previously accumulated interest.

Depreciation — decrease in value over time.

Half-life — time required for half of a quantity to decay.

Asymptote — line that a graph approaches increasingly closely.

Horizontal asymptote — horizontal line approached by a graph.

Interpolation — prediction within the range of known data.

Extrapolation — prediction beyond the range of known data.

Exponential model — exponential function used to represent a real situation.

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Key Takeaways

• Exponential functions have the variable in the exponent.
• A common form is y = abˣ.
• The value a represents the initial amount.
• The value b represents the growth or decay factor.
• If b > 1, the function represents exponential growth.
• If 0 < b < 1, the function represents exponential decay.
• Exponential growth involves repeated multiplication by a factor greater than 1.
• Exponential decay involves repeated multiplication by a factor between 0 and 1.
• Linear relationships have constant differences.
• Exponential relationships have constant ratios for equally spaced inputs.
• Quadratic relationships have constant second differences for equally spaced inputs.
• Linear growth repeatedly adds the same amount.
• Exponential growth repeatedly multiplies by the same factor.
• Percentage growth produces exponential relationships when the percentage is repeatedly applied.
• A growth rate r gives a factor of 1 + r.
• A decay rate r gives a factor of 1 − r.
• A 20% increase corresponds to a factor of 1.20.
• A 20% decrease corresponds to a factor of 0.80.
• Exponential growth graphs become increasingly steep.
• Exponential decay graphs decrease rapidly at first and then flatten.
• Basic exponential growth and decay functions commonly have a horizontal asymptote at y = 0.
• Exponential growth can eventually exceed linear growth dramatically.
• Exponential relationships occur in population growth, bacterial growth, compound interest, inflation, and other situations.
• Exponential decay occurs in radioactive decay, depreciation, medication elimination, and other processes.
• Real-world exponential models are usually valid only over an appropriate range.
• Equations, tables, graphs, and context can all help identify exponential relationships.

Check Your Understanding

1. What feature of an equation usually identifies it as exponential?
2. Which is exponential: y = x², y = 2x, or y = 2ˣ? Explain.
3. In y = 300(1.4ˣ), identify the initial value and growth factor.
4. Does y = 500(0.75ˣ) represent growth or decay? Explain.
5. A population increases by 12% each year. What is its growth factor?
6. A machine loses 8% of its value each year. What is its decay factor?
7. Explain the difference between a growth rate and a growth factor.
8. Determine whether 5, 10, 20, 40, 80 represents a linear or exponential pattern. Explain.
9. Determine whether 5, 10, 15, 20, 25 represents a linear or exponential pattern. Explain.
10. Determine whether 160, 80, 40, 20, 10 represents exponential growth or decay and identify the factor.
11. Compare the meaning of constant difference and constant ratio.
12. A bacteria culture begins with 400 bacteria and doubles every hour. Write an exponential model for the population after t hours.
13. Use your model from Question 12 to predict the population after 6 hours.
14. An investment of $2,000 increases by 5% annually. Write an exponential model for its value after t years.
15. A car worth $40,000 loses 12% of its value each year. Write an exponential model for its value.
16. Explain why compound interest is exponential rather than linear.
17. Explain how the graph of exponential growth differs from a straight-line graph.
18. Explain why an exponential decay graph can approach zero without reaching zero in the basic model.
19. Compare A = 100(1.1ˣ) and B = 100(1.25ˣ). Which begins larger? Which grows faster? Explain.
20. Give one real-world example of exponential growth and one example of exponential decay. For each, explain what is repeatedly multiplied, identify a possible growth or decay factor, and describe one reason the exponential model might eventually stop being appropriate.