SAT Exponential Functions: Growth, Decay, Formulas & Easy Tricks

𝙎𝘼𝙏 𝙀𝙭𝙥𝙤𝙣𝙚𝙣𝙩𝙞𝙖𝙡 𝙁𝙪𝙣𝙘𝙩𝙞𝙤𝙣𝙨 𝙂𝙪𝙞𝙙𝙚

𝙂𝙧𝙤𝙬𝙩𝙝, 𝘿𝙚𝙘𝙖𝙮, 𝙋𝙚𝙧𝙘𝙚𝙣𝙩𝙖𝙜𝙚𝙨, 𝙂𝙧𝙖𝙥𝙝𝙨, 𝙏𝙖𝙗𝙡𝙚𝙨 𝙖𝙣𝙙 𝙒𝙤𝙧𝙙 𝙋𝙧𝙤𝙗𝙡𝙚𝙢𝙨

An exponential function can look complicated at first.

But underneath the equation, table, graph or word problem, there is usually one simple idea:

𝙏𝙝𝙚 𝙨𝙖𝙢𝙚 𝙢𝙪𝙡𝙩𝙞𝙥𝙡𝙞𝙚𝙧 𝙞𝙨 𝙖𝙥𝙥𝙡𝙞𝙚𝙙 𝙖𝙜𝙖𝙞𝙣 𝙖𝙣𝙙 𝙖𝙜𝙖𝙞𝙣.

That single idea connects exponential equations, exponential growth, exponential decay, percentage changes, doubling, halving, tables and graphs.

This guide brings those ideas together in one place.

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✦ 𝟭. 𝙒𝙝𝙖𝙩 𝙄𝙨 𝘼𝙣 𝙀𝙭𝙥𝙤𝙣𝙚𝙣𝙩𝙞𝙖𝙡 𝙁𝙪𝙣𝙘𝙩𝙞𝙤𝙣?

A common exponential function is written as:

𝒇(𝒙) = 𝒂(𝒃ˣ)

There are three important parts.

𝒂 = starting value

𝒃 = multiplication factor

𝒙 = number of repeated changes

The most important clue is that the variable appears in the exponent.

For example:

𝒇(𝒙) = 𝟱(𝟮ˣ)

is exponential because 𝒙 is in the exponent.

But:

𝒇(𝒙) = 𝟱𝒙²

is not an exponential function.

Here, the variable is the base and the exponent is fixed.

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✦ 𝟮. 𝙏𝙝𝙚 𝙈𝙖𝙞𝙣 𝙄𝙙𝙚a: 𝘼𝙙𝙙 𝙊𝙧 𝙈𝙪𝙡𝙩𝙞𝙥𝙡𝙮?

This is one of the quickest ways to distinguish linear and exponential patterns.

Consider:

𝟯, 𝟲, 𝟵, 𝟭𝟮, 𝟭𝟱

The same amount is added each time:

+𝟯

This is a linear pattern.

Now consider:

𝟯, 𝟲, 𝟭𝟮, 𝟮𝟰, 𝟰𝟴

Each value is multiplied by:

×𝟮

This is an exponential pattern.

𝙍𝙚𝙢𝙚𝙢𝙗𝙚𝙧:

𝙎𝙖𝙢𝙚 𝙙𝙞𝙛𝙛𝙚𝙧𝙚𝙣𝙘𝙚 → 𝙡𝙞𝙣𝙚𝙖𝙧

𝙎𝙖𝙢𝙚 𝙧𝙖𝙩𝙞𝙤 → 𝙚𝙭𝙥𝙤𝙣𝙚𝙣𝙩𝙞𝙖𝙡

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✦ 𝟯. 𝙒𝙝𝙖𝙩 𝘿𝙤𝙚𝙨 𝒂 𝙈𝙚𝙖𝙣?

Look at:

𝒇(𝒙) = 𝟴(𝟯ˣ)

The starting value is:

𝒂 = 𝟴

Why?

Set:

𝒙 = 𝟬

Then:

𝒇(𝟬) = 𝟴(𝟯⁰)

Since:

𝟯⁰ = 𝟭

we get:

𝒇(𝟬) = 𝟴

So in:

𝒇(𝒙) = 𝒂(𝒃ˣ)

the value of 𝒂 is the output when 𝒙 = 𝟬.

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✦ 𝟰. 𝙒𝙝𝙖𝙩 𝘿𝙤𝙚𝙨 𝒃 𝙈𝙚𝙖𝙣?

In:

𝒇(𝒙) = 𝒂(𝒃ˣ)

the number 𝒃 tells you how the output changes when 𝒙 increases by 1.

For example:

𝒇(𝒙) = 𝟱(𝟮ˣ)

Values include:

𝒇(𝟬) = 𝟱

𝒇(𝟭) = 𝟭𝟬

𝒇(𝟮) = 𝟮𝟬

𝒇(𝟯) = 𝟰𝟬

Every step multiplies the previous value by:

×𝟮

So the base is the repeated multiplier.

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✦ 𝟱. 𝙂𝙧𝙤𝙬𝙩𝙝 𝙑𝙚𝙧𝙨𝙪𝙨 𝘿𝙚𝙘𝙖𝙮

The base gives you an immediate clue.

𝙄𝙛 𝒃 > 𝟭:

The function shows exponential growth.

Example:

𝒇(𝒙) = 𝟭𝟬(𝟭.𝟮ˣ)

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𝙄𝙛 𝟬 < 𝒃 < 𝟭:

The function shows exponential decay.

Example:

𝒇(𝒙) = 𝟭𝟬(𝟬.𝟴ˣ)

The values get smaller as 𝒙 increases.

𝙌𝙪𝙞𝙘𝙠 𝙘𝙝𝙚𝙘𝙠:

𝒃 > 𝟭 → 𝙂𝙧𝙤𝙬𝙩𝙝

𝟬 < 𝒃 < 𝟭 → 𝘿𝙚𝙘𝙖𝙮

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✦ 𝟲. 𝙏𝙪𝙧𝙣𝙞𝙣𝙜 𝙋𝙚𝙧𝙘𝙚𝙣𝙩𝙖𝙜𝙚𝙨 𝙄𝙣𝙩𝙤 𝙈𝙪𝙡𝙩𝙞𝙥𝙡𝙞𝙚𝙧𝙨

This is one of the most important skills in exponential word problems.

Suppose something increases by 𝟭𝟬%.

A 10% increase means the new amount is:

𝟭𝟬𝟬% + 𝟭𝟬% = 𝟭𝟭𝟬%

As a decimal:

𝟭.𝟭𝟬

Therefore:

𝟭𝟬% 𝙞𝙣𝙘𝙧𝙚𝙖𝙨𝙚 → ×𝟭.𝟭𝟬

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Suppose something increases by 𝟮𝟱%.

𝟭𝟬𝟬% + 𝟮𝟱% = 𝟭𝟮𝟱%

Therefore:

𝟮𝟱% 𝙞𝙣𝙘𝙧𝙚𝙖𝙨𝙚 → ×𝟭.𝟮𝟱

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✦ 𝟳. 𝙋𝙚𝙧𝙘𝙚𝙣𝙩𝙖𝙜𝙚 𝘿𝙚𝙘𝙧𝙚𝙖𝙨𝙚𝙨

Suppose something decreases by 𝟭𝟬%.

The amount remaining is:

𝟭𝟬𝟬% − 𝟭𝟬% = 𝟵𝟬%

As a decimal:

𝟬.𝟵𝟬

Therefore:

𝟭𝟬% 𝙙𝙚𝙘𝙧𝙚𝙖𝙨𝙚 → ×𝟬.𝟵𝟬

Similarly:

𝟮𝟬% decrease → ×𝟬.𝟴𝟬

𝟯𝟬% decrease → ×𝟬.𝟳𝟬

𝟰𝟬% decrease → ×𝟬.𝟲𝟬

𝟱𝟬% decrease → ×𝟬.𝟱𝟬

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⚠️ ✦ 𝟴. 𝙏𝙝𝙚 𝘽𝙞𝙜 𝙋𝙚𝙧𝙘𝙚𝙣𝙩𝙖𝙜𝙚 𝙏𝙧𝙖𝙥

Suppose a quantity decreases by 𝟴𝟬%.

The incorrect multiplier is:

𝟬.𝟴𝟬

Why?

Because 80% is the amount removed, not the amount remaining.

The amount remaining is:

𝟭𝟬𝟬% − 𝟴𝟬% = 𝟮𝟬%

Therefore:

𝟴𝟬% 𝙙𝙚𝙘𝙧𝙚𝙖𝙨𝙚 → ×𝟬.𝟮𝟬

This is an easy place to lose a question.

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✦ 𝟵. 𝘽𝙪𝙞𝙡𝙙𝙞𝙣𝙜 𝘼𝙣 𝙀𝙭𝙥𝙤𝙣𝙚𝙣𝙩𝙞𝙖𝙡 𝙈𝙤𝙙𝙚𝙡

Suppose a population begins at:

𝟮𝟬𝟬𝟬

and increases by:

𝟱% per year

Starting value:

𝒂 = 𝟮𝟬𝟬𝟬

Growth multiplier:

𝟭 + 𝟬.𝟬𝟱 = 𝟭.𝟬𝟱

Therefore:

𝑷(𝒕) = 𝟮𝟬𝟬𝟬(𝟭.𝟬𝟱ᵗ)

The structure is always:

𝙎𝙩𝙖𝙧𝙩𝙞𝙣𝙜 𝙫𝙖𝙡𝙪𝙚 × (𝙜𝙧𝙤𝙬𝙩𝙝 𝙛𝙖𝙘𝙩𝙤𝙧)ᵗ

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✦ 𝟭𝟬. 𝘿𝙚𝙘𝙖𝙮 𝙈𝙤𝙙𝙚𝙡𝙨

Suppose a machine is worth:

$𝟭𝟱𝟬𝟬𝟬

and loses:

𝟭𝟮% of its value each year

The amount remaining each year is:

𝟭 − 𝟬.𝟭𝟮 = 𝟬.𝟴𝟴

Therefore:

𝑽(𝒕) = 𝟭𝟱𝟬𝟬𝟬(𝟬.𝟴𝟴ᵗ)

Notice something important.

The machine does not lose $1,800 every year.

It loses 12% of its current value.

That distinction creates exponential decay.

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✦ 𝟭𝟭. 𝙀𝙭𝙥𝙤𝙣𝙚𝙣𝙩𝙞𝙖𝙡 𝙂𝙧𝙤𝙬𝙩𝙝 𝙑𝙨 𝙇𝙞𝙣𝙚𝙖𝙧 𝙂𝙧𝙤𝙬𝙩𝙝

Suppose two quantities start at 100.

𝙇𝙞𝙣𝙚𝙖𝙧

Increase by 20 each time:

𝟭𝟬𝟬, 𝟭𝟮𝟬, 𝟭𝟰𝟬, 𝟭𝟲𝟬, 𝟭𝟴𝟬

𝙀𝙭𝙥𝙤𝙣𝙚𝙣𝙩𝙞𝙖𝙡

Increase by 20% each time:

𝟭𝟬𝟬, 𝟭𝟮𝟬, 𝟭𝟰𝟰, 𝟭𝟳𝟮.𝟴, 𝟮𝟬𝟳.𝟯𝟲

The first adds the same amount.

The second multiplies by the same factor.

𝙏𝙝𝙖𝙩 𝙞𝙨 𝙩𝙝𝙚 𝙚𝙨𝙨𝙨𝙚𝙣𝙩𝙞𝙖𝙡 𝙙𝙞𝙛𝙛𝙚𝙧𝙚𝙣𝙘𝙚.

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✦ 𝟭𝟮. 𝙁𝙞𝙣𝙙𝙞𝙣𝙜 𝙖𝙣 𝙀𝙭𝙥𝙤𝙣𝙚𝙣𝙩𝙞𝙖𝙡 𝙁𝙪𝙣𝙘𝙩𝙞𝙤𝙣 𝙁𝙧𝙤𝙢 𝙖 𝙏𝙖𝙗𝙡𝙚

Consider:

𝒙	𝒇(𝒙)
𝟬	𝟱
𝟭	𝟭𝟬
𝟮	𝟮𝟬
𝟯	𝟰𝟬
𝟰	𝟴𝟬

Look at consecutive ratios.

𝟭𝟬 ÷ 𝟱 = 𝟮

𝟮𝟬 ÷ 𝟭𝟬 = 𝟮

𝟰𝟬 ÷ 𝟮𝟬 = 𝟮

𝟴𝟬 ÷ 𝟰𝟬 = 𝟮

The multiplier is:

𝒃 = 𝟮

The starting value is:

𝒂 = 𝟱

Therefore:

𝒇(𝒙) = 𝟱(𝟮ˣ)

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✦ 𝟭𝟯. 𝙁𝙞𝙣𝙙𝙞𝙣𝙜 𝙖 𝙈𝙞𝙨𝙨𝙞𝙣𝙜 𝙑𝙖𝙡𝙪𝙚

Suppose:

𝒙	𝒇(𝒙)
𝟬	𝟲
𝟭	𝟭𝟴
𝟮	?
𝟯	𝟭𝟲𝟮

The multiplier is:

𝟭𝟴 ÷ 𝟲 = 𝟯

So:

𝟲 × 𝟯 = 𝟭𝟴

𝟭𝟴 × 𝟯 = 𝟱𝟰

𝟱𝟰 × 𝟯 = 𝟭𝟲𝟮

Therefore:

𝒇(𝟮) = 𝟱𝟰

You do not always need to build the entire equation.

Sometimes the pattern is enough.

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✦ 𝟭𝟰. 𝘿𝙤𝙪𝙗𝙡𝙞𝙣𝙜 𝙋𝙖𝙩𝙩𝙚𝙧𝙣𝙨

Suppose a quantity doubles every 3 hours.

If 𝒕 represents hours, the model is:

𝑨(𝒕) = 𝑨₀(𝟮)ᵗ⁄³

Why is the exponent 𝒕⁄𝟯?

Because one doubling occurs every 3 hours.

For example, if the starting amount is 100:

After 3 hours:

𝟭𝟬𝟬 × 𝟮 = 𝟮𝟬𝟬

After 6 hours:

𝟭𝟬𝟬 × 𝟮² = 𝟰𝟬𝟬

After 9 hours:

𝟭𝟬𝟬 × 𝟮³ = 𝟴𝟬𝟬

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✦ 𝟭𝟱. 𝙏𝙧𝙞𝙥𝙡𝙞𝙣𝙜 𝙋𝙖𝙩𝙩𝙚𝙧𝙣𝙨

If a quantity triples every 4 hours:

𝑨(𝒕) = 𝑨₀(𝟯)ᵗ⁄⁴

If the initial value is 50:

𝑨(𝒕) = 𝟱𝟬(𝟯)ᵗ⁄⁴

After 4 hours:

𝟱𝟬 × 𝟯 = 𝟭𝟱𝟬

After 8 hours:

𝟱𝟬 × 𝟯² = 𝟰𝟱𝟬

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✦ 𝟭𝟲. 𝙃𝙖𝙡𝙛-𝙇𝙞𝙛𝙚 𝙋𝙖𝙩𝙩𝙚𝙧𝙣𝙨

If a quantity is reduced to half every 5 years:

𝑨(𝒕) = 𝑨₀(𝟭⁄𝟮)ᵗ⁄⁵

Suppose:

𝑨₀ = 𝟭𝟲𝟬

Then:

After 5 years:

𝟭𝟲𝟬 × 𝟭⁄𝟮 = 𝟴𝟬

After 10 years:

𝟭𝟲𝟬 × (𝟭⁄𝟮)² = 𝟰𝟬

After 15 years:

𝟭𝟲𝟬 × (𝟭⁄𝟮)³ = 𝟮𝟬

The quantity keeps being multiplied by the same factor.

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✦ 𝟭𝟳. 𝙒𝙝𝙚𝙣 𝙏𝙝𝙚 𝙏𝙞𝙢𝙚 𝙐𝙣𝙞𝙩 𝘾𝙝𝙖𝙣𝙜𝙚𝙨

Be careful when the time unit in the question does not match the time unit in the model.

Suppose a quantity doubles every:

4 years

and 𝒕 is measured in years.

Then:

𝑨(𝒕) = 𝑨₀(𝟮)ᵗ⁄⁴

But if 𝒕 represents four-year periods instead, the model could simply be:

𝑨(𝒕) = 𝑨₀(𝟮ᵗ)

Always ask:

“𝙒𝙝𝙖𝙩 𝙙𝙤𝙚𝙨 𝟭 𝙪𝙣𝙞𝙩 𝙤𝙛 𝒙 𝙧𝙚𝙥𝙧𝙚𝙨𝙚𝙣𝙩?”

That one question can prevent a major modeling error.

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✦ 𝟭𝟴. 𝙀𝙭𝙥𝙤𝙣𝙚𝙣𝙩 𝙍𝙪𝙡𝙚𝙨

Exponential functions become much easier when the basic exponent rules are automatic.

𝒂⁰ = 𝟭

𝒂ᵐ × 𝒂ⁿ = 𝒂ᵐ⁺ⁿ

𝒂ᵐ ÷ 𝒂ⁿ = 𝒂ᵐ⁻ⁿ

(𝒂ᵐ)ⁿ = 𝒂ᵐⁿ

𝒂⁻ⁿ = 𝟭⁄𝒂ⁿ

For example:

𝟮³ × 𝟮⁴ = 𝟮⁷

because:

𝟯 + 𝟰 = 𝟳

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⚠️ ✦ 𝟭𝟵. 𝘿𝙤 𝙉𝙤𝙩 𝘼𝙙𝙙 𝙀𝙭𝙥𝙤𝙣𝙚𝙣𝙩𝙨 𝙒𝙝𝙚𝙣 𝘼𝙙𝙙𝙞𝙣𝙜

This rule:

𝒂ᵐ × 𝒂ⁿ = 𝒂ᵐ⁺ⁿ

is for multiplication.

It does not mean:

𝒂ᵐ + 𝒂ⁿ = 𝒂ᵐ⁺ⁿ

For example:

𝟮² + 𝟮³

equals:

𝟰 + 𝟴 = 𝟭𝟮

It does not equal:

𝟮⁵

Always look at the operation before choosing an exponent rule.

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✦ 𝟮𝟬. 𝙎𝙤𝙡𝙫𝙞𝙣𝙜 𝙎𝙞𝙢𝙥𝙡𝙚 𝙀𝙭𝙥𝙤𝙣𝙚𝙣𝙩𝙞𝙖𝙡 𝙀𝙦𝙪𝙖𝙩𝙞𝙤𝙣𝙨

Consider:

𝟮ˣ = 𝟯𝟮

Rewrite 32 as a power of 2:

𝟯𝟮 = 𝟮⁵

Therefore:

𝟮ˣ = 𝟮⁵

So:

𝒙 = 𝟱

The key strategy is:

𝙏𝙧𝙮 𝙩𝙤 𝙬𝙧𝙞𝙩𝙚 𝙗𝙤𝙩𝙝 𝙨𝙞𝙙𝙚𝙨 𝙬𝙞𝙩𝙝 𝙩𝙝𝙚 𝙨𝙖𝙢𝙚 𝙗𝙖𝙨𝙚.

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✦ 𝟮𝟭. 𝙂𝙧𝙖𝙥𝙝𝙨 𝙊𝙛 𝙀𝙭𝙥𝙤𝙣𝙚𝙣𝙩𝙞𝙖𝙡 𝙁𝙪𝙣𝙘𝙩𝙞𝙤𝙣𝙨

For:

𝒇(𝒙) = 𝒂(𝒃ˣ)

the graph is curved rather than a straight line.

If:

𝒃 > 𝟭

the graph rises as 𝒙 increases.

If:

𝟬 < 𝒃 < 𝟭

the graph falls as 𝒙 increases.

The graph passes through:

(𝟬, 𝒂)

because:

𝒇(𝟬) = 𝒂

For the basic form with no vertical shift, the graph approaches:

𝒚 = 𝟬

as the curve extends in the appropriate direction.

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✦ 𝟮𝟮. 𝙒𝙝𝙖𝙩 𝘿𝙤𝙚𝙨 𝒌 𝘿𝙤?

Consider:

𝒇(𝒙) = 𝒂(𝒃ˣ) + 𝒌

The +𝒌 moves the entire graph vertically.

For example:

𝒇(𝒙) = 𝟯(𝟮ˣ) + 𝟱

has horizontal asymptote:

𝒚 = 𝟱

The vertical shift changes the long-term position of the graph.

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✦ 𝟮𝟯. 𝙁𝙞𝙣𝙙𝙞𝙣𝙜 𝙏𝙝𝙚 𝙂𝙧𝙤𝙬𝙩𝙝 𝙁𝙖𝙘𝙩𝙤𝙧

Suppose a quantity changes from:

𝟮𝟬𝟬 → 𝟮𝟯𝟬

The multiplier is:

𝟮𝟯𝟬 ÷ 𝟮𝟬𝟬 = 𝟭.𝟭𝟱

Therefore the growth factor is:

𝟭.𝟭𝟱

The percentage increase is:

𝟭.𝟭𝟱 − 𝟭 = 𝟬.𝟭𝟱

which is:

𝟭𝟱%

So:

×𝟭.𝟭𝟱 = 𝟭𝟱% growth

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✦ 𝟮𝟰. 𝙁𝙞𝙣𝙙𝙞𝙣𝙜 𝙏𝙝𝙚 𝘿𝙚𝙘𝙖𝙮 𝙍𝙖𝙩𝙚

Suppose a quantity changes from:

𝟱𝟬𝟬 → 𝟰𝟮𝟱

The multiplier is:

𝟰𝟮𝟱 ÷ 𝟱𝟬𝟬 = 𝟬.𝟴𝟱

The amount remaining is:

𝟴𝟱%

Therefore the decrease is:

𝟭𝟱%

So:

×𝟬.𝟴𝟱 = 𝟭𝟱% decay

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✦ 𝟮𝟱. 𝙏𝙝𝙚 𝙊𝙣𝙚-𝙎𝙩𝙚𝙥 𝙈𝙪𝙡𝙩𝙞𝙥𝙡𝙞𝙚𝙧 𝙍𝙪𝙡𝙚

For:

𝒇(𝒙) = 𝒂(𝒃ˣ)

we can write:

𝒇(𝒙 + 𝟭) = 𝒃𝒇(𝒙)

This is powerful because it tells you exactly what happens after one additional step.

Suppose:

𝒇(𝒙 + 𝟭) = 𝟰𝒇(𝒙)

Then the multiplier is:

𝒃 = 𝟰

If:

𝒇(𝟬) = 𝟯

then:

𝒇(𝟭) = 𝟭𝟮

𝒇(𝟮) = 𝟰𝟴

𝒇(𝟯) = 𝟭𝟵𝟮

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✦ 𝟮𝟲. 𝙎𝘼𝙏 𝙒𝙤𝙧𝙙 𝙋𝙧𝙤𝙗𝙡𝙚𝙢 𝙏𝙧𝙖𝙣𝙨𝙡𝙖𝙩𝙞𝙤𝙣

Words such as these should immediately make you think about exponential models:

“increases by 𝟱% each year”

→ ×𝟭.𝟬𝟱

“decreases by 𝟭𝟮% each month”

→ ×𝟬.𝟴𝟴

“doubles every 𝟯 hours”

→ ×𝟮 every 𝟯 hours

“triples every 𝟱 days”

→ ×𝟯 every 𝟱 days

“is reduced by half every 𝟰 years”

→ ×𝟭⁄𝟮 every 𝟰 years

The wording changes.

The underlying mathematics remains the same.

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✦ 𝟮𝟳. 𝙒𝙤𝙧𝙙 𝙋𝙧𝙤𝙗𝙡𝙚𝙢: 𝙂𝙧𝙤𝙬𝙩𝙝

A town has a population of 𝟭𝟬,𝟬𝟬𝟬 and grows by 𝟮% each year.

Step 𝟭: Starting value

𝒂 = 𝟭𝟬𝟬𝟬𝟬

Step 𝟮: Growth factor

𝟭 + 𝟬.𝟬𝟮 = 𝟭.𝟬𝟮

Step 𝟯: Build the model

𝑷(𝒕) = 𝟭𝟬𝟬𝟬𝟬(𝟭.𝟬𝟮ᵗ)

The equation describes the population after 𝒕 years.

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✦ 𝟮𝟴. 𝙒𝙤𝙧𝙙 𝙋𝙧𝙤𝙗𝙡𝙚𝙢: 𝘿𝙚𝙘𝙖𝙮

A car is worth $𝟮𝟬,𝟬𝟬𝟬 and loses 𝟭𝟱% of its value each year.

Remaining percentage:

𝟭𝟬𝟬% − 𝟭𝟱% = 𝟴𝟱%

Multiplier:

𝟬.𝟴𝟱

Therefore:

𝑽(𝒕) = 𝟮𝟬𝟬𝟬𝟬(𝟬.𝟴𝟱ᵗ)

Notice that the exponent counts the number of years.

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✦ 𝟮𝟵. 𝙏𝙝𝙚 𝙎𝙖𝙢𝙚 𝙋𝙚𝙧𝙘𝙚𝙣𝙩𝙖𝙜𝙚 𝘿𝙤𝙚𝙨 𝙉𝙤𝙩 𝙈𝙚𝙖𝙣 𝙏𝙝𝙚 𝙎𝙖𝙢𝙚 𝘼𝙢𝙤𝙪𝙣𝙩

This is a crucial concept.

Suppose a value is:

𝟭𝟬𝟬

and decreases by 𝟭𝟬%.

First decrease:

𝟭𝟬𝟬 × 𝟬.𝟵 = 𝟵𝟬

Second decrease:

𝟵𝟬 × 𝟬.𝟵 = 𝟴𝟭

Third decrease:

𝟴𝟭 × 𝟬.𝟵 = 𝟳𝟮.𝟵

The decrease amounts are:

𝟭𝟬

then:

𝟵

then:

𝟴.𝟭

The percentage remains the same.

The actual amount changes.

That is why the process is exponential.

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✦ 𝟯𝟬. 𝙒𝙝𝙮 𝙀𝙭𝙥𝙤𝙣𝙚𝙣𝙩𝙞𝙖𝙡 𝙂𝙧𝙤𝙬𝙩𝙝 𝘾𝙖𝙣 𝙎𝙪𝙧𝙥𝙖𝙨𝙨 𝙇𝙞𝙣𝙚𝙖𝙧 𝙂𝙧𝙤𝙬𝙩𝙝

Imagine:

Linear: add 10 each step.

Exponential: multiply by 1.10 each step.

Starting from 100:

Linear:

𝟭𝟬𝟬 → 𝟭𝟭𝟬 → 𝟭𝟮𝟬 → 𝟭𝟯𝟬 → 𝟭𝟰𝟬

Exponential:

𝟭𝟬𝟬 → 𝟭𝟭𝟬 → 𝟭𝟮𝟭 → 𝟭𝟯𝟯.𝟭 → 𝟭𝟰𝟲.𝟰𝟭

At first the values look similar.

But repeated multiplication can eventually produce a very large difference.

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✦ 𝟯𝟭. 𝙀𝙭𝙥𝙤𝙣𝙚𝙣𝙩𝙞𝙖𝙡 𝙂𝙧𝙤𝙬𝙩𝙝 𝘾𝙖𝙣 𝘽𝙚 𝙁𝙖𝙨𝙩

Consider:

𝒇(𝒙) = 𝟮(𝟯ˣ)

The first few values are:

𝒙 = 𝟬 → 𝟮

𝒙 = 𝟭 → 𝟲

𝒙 = 𝟮 → 𝟭𝟴

𝒙 = 𝟯 → 𝟱𝟰

𝒙 = 𝟰 → 𝟭𝟲𝟮

Every step multiplies the previous output by 3.

That repeated multiplication is the heart of exponential growth.

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✦ 𝟯𝟮. 𝙏𝙝𝙚 𝙈𝙤𝙨𝙩 𝘾𝙤𝙢𝙢𝙤𝙣 𝙈𝙞𝙨𝙩𝙖𝙠𝙚𝙨

❌ 𝙈𝙞𝙨𝙩𝙖𝙠𝙚 𝟭: 𝙐𝙨𝙞𝙣𝙜 𝟬.𝟬𝟱 𝙁𝙤𝙧 𝟱% 𝙂𝙧𝙤𝙬𝙩𝙝

Correct:

𝟭.𝟬𝟱

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❌ 𝙈𝙞𝙨𝙩𝙖𝙠𝙚 𝟮: 𝙐𝙨𝙞𝙣𝙜 𝟬.𝟮𝟬 𝙁𝙤𝙧 𝟮𝟬% 𝘿𝙚𝙘𝙧𝙚𝙖𝙨𝙚

Correct:

𝟬.𝟴𝟬

---

❌ 𝙈𝙞𝙨𝙩𝙖𝙠𝙚 𝟯: 𝘾𝙝𝙚𝙘𝙠𝙞𝙣𝙜 𝘿𝙞𝙛𝙛𝙚𝙧𝙚𝙣𝙘𝙚𝙨 𝙄𝙣𝙨𝙩𝙚𝙖𝙙 𝙊𝙛 𝙍𝙖𝙩𝙞𝙤𝙨

For exponential tables, divide consecutive values.

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❌ 𝙈𝙞𝙨𝙩𝙖𝙠𝙚 𝟰: 𝙄𝙜𝙣𝙤𝙧𝙞𝙣𝙜 𝙏𝙞𝙢𝙚 𝙐𝙣𝙞𝙩𝙨

“Doubles every 5 years” does not mean it doubles every year.

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❌ 𝙈𝙞𝙨𝙩𝙖𝙠𝙚 𝟱: 𝙈𝙞𝙭𝙞𝙣𝙜 𝙐𝙥 𝙎𝙩𝙖𝙧𝙩𝙞𝙣𝙜 𝙑𝙖𝙡𝙪𝙚 𝘼𝙣𝙙 𝙂𝙧𝙤𝙬𝙩𝙝 𝙁𝙖𝙘𝙩𝙤𝙧

In:

𝒇(𝒙) = 𝟱𝟬(𝟭.𝟬𝟰ˣ)

50 is the starting value.

1.04 is the growth factor.

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✦ 𝟯𝟯. 𝙁𝙖𝙨𝙩 𝙎𝙖𝙩 𝙎𝙩𝙧𝙖𝙩𝙚𝙜𝙮

When you see an exponential question, stop before calculating.

Ask these questions:

① 𝙒𝙝𝙖𝙩 𝙞𝙨 𝙩𝙝𝙚 𝙨𝙩𝙖𝙧𝙩𝙞𝙣𝙜 𝙫𝙖𝙡𝙪𝙚?

② 𝙒𝙝𝙖𝙩 𝙞𝙨 𝙩𝙝𝙚 𝙢𝙪𝙡𝙩𝙞𝙥𝙡𝙞𝙚𝙧?

③ 𝙄𝙨 𝙞𝙩 𝙜𝙧𝙤𝙬𝙩𝙝 𝙤𝙧 𝙙𝙚𝙘𝙖𝙮?

④ 𝙒𝙝𝙖𝙩 𝙙𝙤𝙚𝙨 𝒙 𝙧𝙚𝙥𝙧𝙚𝙨𝙚𝙣𝙩?

⑤ 𝙃𝙤𝙬 𝙢𝙖𝙣𝙮 𝙩𝙞𝙢𝙚𝙨 𝙞𝙨 𝙩𝙝𝙚 𝙛𝙖𝙘𝙩𝙤𝙧 𝙖𝙥𝙥𝙡𝙞𝙚𝙙?

If you answer those five questions, many apparently difficult problems become much simpler.

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✦ 𝟯𝟰. 𝙌𝙪𝙞𝙘𝙠 𝙎𝘼𝙏 𝙋𝙧𝙖𝙘𝙩𝙞𝙘𝙚

𝙌𝙪𝙚𝙨𝙩𝙞𝙤𝙣 𝟭

Which equation represents exponential growth?

𝘼) 𝒚 = 𝟯𝒙 + 𝟮

𝘽) 𝒚 = 𝟯𝒙² + 𝟮

𝘾) 𝒚 = 𝟯(𝟭.𝟱ˣ)

𝘿) 𝒚 = 𝟯⁄𝒙

𝘼𝙣𝙨𝙬𝙚𝙧: 𝘾

The variable appears in the exponent and the base is greater than 1.

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𝙌𝙪𝙚𝙨𝙩𝙞𝙤𝙣 𝟮

A quantity increases by 12% each year.

What is the growth factor?

𝘼𝙣𝙨𝙬𝙚𝙧: 𝟭.𝟭𝟮

Because:

𝟭 + 𝟬.𝟭𝟮 = 𝟭.𝟭𝟮

---

𝙌𝙪𝙚𝙨𝙩𝙞𝙤𝙣 𝟯

A quantity decreases by 35% each month.

What multiplier should be used?

𝘼𝙣𝙨𝙬𝙚𝙧: 𝟬.𝟲𝟱

Because:

𝟭 − 𝟬.𝟯𝟱 = 𝟬.𝟲𝟱

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𝙌𝙪𝙚𝙨𝙩𝙞𝙤𝙣 𝟰

The values in a table are:

𝟰, 𝟭𝟮, 𝟯𝟲, 𝟭𝟬𝟴

What is the common ratio?

𝘼𝙣𝙨𝙬𝙚𝙧: 𝟯

because:

𝟭𝟮 ÷ 𝟰 = 𝟯

𝟯𝟲 ÷ 𝟭𝟮 = 𝟯

𝟭𝟬𝟴 ÷ 𝟯𝟲 = 𝟯

---

𝙌𝙪𝙚𝙨𝙩𝙞𝙤𝙣 𝟱

A quantity starts at 80 and doubles every 4 hours.

What is its value after 12 hours?

There are:

𝟭𝟮 ÷ 𝟰 = 𝟯

doubling periods.

Therefore:

𝟴𝟬 × 𝟮³

= 𝟴𝟬 × 𝟴

= 𝟲𝟰𝟬

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✦ 𝟯𝟱. 𝙏𝙝𝙚 𝙀𝙭𝙥𝙤𝙣𝙚𝙣𝙩𝙞𝙖𝙡 𝙁𝙪𝙣𝙘𝙩𝙞𝙤𝙣 𝙈𝙚𝙢𝙤𝙧𝙮 𝘾𝙖𝙧𝙙

𝒇(𝒙) = 𝒂(𝒃ˣ)

𝒂 → starting value

𝒃 → repeated multiplier

𝒙 → number of steps

𝒃 > 𝟭 → growth

𝟬 < 𝒃 < 𝟭 → decay

𝒇(𝟬) = 𝒂

𝒓% increase → ×(𝟭 + 𝒓)

𝒓% decrease → ×(𝟭 − 𝒓)

constant difference → linear

constant ratio → exponential

double → ×𝟮

triple → ×𝟯

half → ×𝟭⁄𝟮

𝒇(𝒙 + 𝟭) = 𝒃𝒇(𝒙)

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✦ 𝟯𝟲. 𝙎𝘼𝙏 𝙀𝙭𝙥𝙤𝙣𝙚𝙣𝙩𝙞𝙖𝙡 𝙁𝙪𝙣𝙘𝙩𝙞𝙤𝙣𝙨 𝙁𝘼𝙌

𝙒𝙝𝙖𝙩 𝙞𝙨 𝙖𝙣 𝙚𝙭𝙥𝙤𝙣𝙚𝙣𝙩𝙞𝙖𝙡 𝙛𝙪𝙣𝙘𝙩𝙞𝙤𝙣?

A function in which the variable appears in the exponent, commonly written:

𝒇(𝒙) = 𝒂(𝒃ˣ)

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𝙃𝙤𝙬 𝙙𝙤 𝙄 𝙞𝙙𝙚𝙣𝙩𝙞𝙛𝙮 𝙚𝙭𝙥𝙤𝙣𝙚𝙣𝙩𝙞𝙖𝙡 𝙜𝙧𝙤𝙬𝙩𝙝?

Look at the base.

If:

𝒃 > 𝟭

the function grows.

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𝙃𝙤𝙬 𝙙𝙤 𝙄 𝙞𝙙𝙚𝙣𝙩𝙞𝙛𝙮 𝙚𝙭𝙥𝙤𝙣𝙚𝙣𝙩𝙞𝙖𝙡 𝙙𝙚𝙘𝙖𝙮?

If:

𝟬 < 𝒃 < 𝟭

the function decays.

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𝙒𝙝𝙖𝙩 𝙞𝙨 𝟮𝟬% 𝙜𝙧𝙤𝙬𝙩𝙝 𝙖𝙨 𝙖 𝙛𝙖𝙘𝙩𝙤𝙧?

𝟭.𝟮

---

𝙒𝙝𝙖𝙩 𝙞𝙨 𝟮𝟬% 𝙙𝙚𝙘𝙖𝙮 𝙖𝙨 𝙖 𝙛𝙖𝙘𝙩𝙤𝙧?

𝟬.𝟴

---

𝙃𝙤𝙬 𝙙𝙤 𝙄 𝙛𝙞𝙣𝙙 𝙖𝙣 𝙚𝙭𝙥𝙤𝙣𝙚𝙣𝙩𝙞𝙖𝙡 𝙥𝙖𝙩𝙩𝙚𝙧𝙣 𝙞𝙣 𝙖 𝙩𝙖𝙗𝙡𝙚?

Divide consecutive output values.

If the ratios remain the same, the pattern is exponential.

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𝙒𝙝𝙖𝙩 𝙞𝙨 𝙩𝙝𝙚 𝙙𝙞𝙛𝙛𝙚𝙧𝙚𝙣𝙘𝙚 𝙗𝙚𝙩𝙬𝙚𝙚𝙣 𝙚𝙭𝙥𝙤𝙣𝙚𝙣𝙩𝙞𝙖𝙡 𝙖𝙣𝙙 𝙡𝙞𝙣𝙚𝙖𝙧 𝙜𝙧𝙤𝙬𝙩𝙝?

Linear growth repeatedly adds the same amount.

Exponential growth repeatedly multiplies by the same factor.

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𝙒𝙝𝙖𝙩 𝙙𝙤𝙚𝙨 𝒂 𝙢𝙚𝙖𝙣 𝙞𝙣 𝒇(𝒙) = 𝒂(𝒃ˣ)?

It is the starting value because:

𝒇(𝟬) = 𝒂

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𝙒𝙝𝙖𝙩 𝙙𝙤𝙚𝙨 𝒃 𝙢𝙚𝙖𝙣?

It is the multiplier applied whenever 𝒙 increases by one unit.

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✦ 𝙁𝙞𝙣𝙖𝙡 𝙏𝙖𝙠𝙚𝙖𝙬𝙖𝙮

The easiest way to understand exponential functions is not to memorize dozens of separate examples.

Understand the pattern:

𝙇𝙞𝙣𝙚𝙖𝙧 → 𝙖𝙙𝙙 𝙩𝙝𝙚 𝙨𝙖𝙢𝙚 𝙖𝙢𝙤𝙪𝙣𝙩.

𝙀𝙭𝙥𝙤𝙣𝙚𝙣𝙩𝙞𝙖𝙡 → 𝙢𝙪𝙡𝙩𝙞𝙥𝙡𝙮 𝙗𝙮 𝙩𝙝𝙚 𝙨𝙖𝙢𝙚 𝙛𝙖𝙘𝙩𝙤𝙧.

When you see an exponential question, find:

𝙩𝙝𝙚 𝙨𝙩𝙖𝙧𝙩𝙞𝙣𝙜 𝙫𝙖𝙡𝙪𝙚

𝙩𝙝𝙚 𝙢𝙪𝙡𝙩𝙞𝙥𝙡𝙞𝙚𝙧

𝙩𝙝𝙚 𝙩𝙞𝙢𝙚 𝙞𝙣𝙩𝙚𝙧𝙫𝙖𝙡

𝙖𝙣𝙙 𝙩𝙝𝙚 𝙣𝙪𝙢𝙗𝙚𝙧 𝙤𝙛 𝙧𝙚𝙥𝙚𝙖𝙩𝙚𝙙 𝙘𝙝𝙖𝙣𝙜𝙚𝙨.

Once those four pieces are clear, the equation usually becomes much easier to see.

𝙎𝙚𝙚 𝙩𝙝𝙚 𝙛𝙖𝙘𝙩𝙤𝙧. 𝙍𝙚𝙘𝙤𝙜𝙣𝙞𝙯𝙚 𝙩𝙝𝙚 𝙥𝙖𝙩𝙩𝙚𝙧𝙣. 𝙏𝙝𝙚𝙣 𝙨𝙤𝙡𝙫𝙚.

other pages to explore

SAT MATH FORMULA  SHEET FOR QUICK REFERENCE

SAT MATH FORMULA CHEAT SHEET 

ALGEBRA

SO;VING LINEAR EQUATIONS [PART 1]

LINEAR EQUATIONS [PART 1]

SOLVING LINEAR EQUATIONS [PART II]

LINEAR EQUATIONS [PART II]

SYSTEM OF EQUATIONS  [ PART  I ]

system of equations part I

SAT Systems of Equations: Part 2 — Hard Questions, Word Problems, Graphs, Parameters & SAT Tricks

SAT system of equations 2

 PARAMETER QUESTIONS IN SYSTEMS OF LINEAR EQUATIONS

to find the parameter, unique solution, no solution infinite number of solutions

QUADRATIC EQUATIONS [PART I] 

QUADRATIC EQUATIONS [PART I] 

QUADRATIC EQUATIONS [PART II] 

QUADRATIC EQUATIONS [PART II] 

Linear Inequality

linear inequalities

PERCENTAGES

PERCENTAGES [introduction]

PERCENTAGES [part 1]

PERCENTAGE INCREASE AND DECREASE

PERCENTAGES [part 2]

SUCCESSIVE PERCENTAGES DISCOUNT PROFIT LOSS

PERCENTAGES [part3]

Percentages in Data Analysis, Graphs, Tables, Probability,SAT Geometry Notes:  Study Guide with Formulas, Tricks, Practice Questions and Advanced Word Problems

PERCENTAGES [part4]

SAT Reverse Percentage Problems

Reverse percentage problems for SAT

GEOMETRY

SAT Geometry Notes:  Study Guide with Formulas, Tricks, Practice Questions

geometry notes  [ part 1 ] 

SAT Word Problems Made Easy: Tricks to Solve Digital SAT Math Problems Faster

SAT Word Problems Made Easy

The Big Word-Problem Families:  SAT Math Guide

Big Word-Problem Families

How SAT Word Problems Hide the Equation: Skill That Makes Difficult Questions Easier

How SAT word problems hide equations

SAT Word Problems Distance, Speed, Work, Age, Mixtures & Profit — A Smarter Way to Decode the Question

SAT Word Problems Made Easy

SAT Exponential Functions: Growth, Decay, Formulas & Easy Tricks

SAT Exponential Functions

SAT Function Transformations: A Simple Way to Master Shifts, Reflections, Stretches and Compressions

SAT function transformations tips and tricks

SAT Probability Made Easy: Conditional Probability, Tables & Tricky Questions

SAT probability guide

SAT Conditional Probability

Conditional Probability