Exponential Functions

Q: How can I study Exponential Functions effectively?

Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 10 problems provided, checking solutions as you go. Regular review and active practice are key to retention.

Q: Is this Exponential Functions study guide free?

Yes — all study notes, flashcards, and practice problems for Exponential Functions on Study Mondo are free to access. No account is needed.

Q: What course covers Exponential Functions?

Exponential Functions is part of the SAT Prep course on Study Mondo, specifically in the Advanced Math section. You can explore the full course for more related topics and practice resources.

Q: Are there practice problems for Exponential Functions?

Yes, this page includes 10 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Master exponential growth and decay, interpret exponential expressions, solve exponential equations, and model real-world phenomena with exponential functions.

Written and reviewed by the Study Mondo Education TeamLast updated April 18, 2026

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Exponential Functions and Equations on the SAT

What Is an Exponential Function?

An exponential function has the form: $f(x) = a \cdot b^x$

Where:

$a$ = initial value (when $x = 0$ )
$b$ = base (growth/decay factor)
$x$ = typically time or number of periods

Growth vs. Decay

Condition	Type	Example
$b > 1$	Exponential Growth	Population doubling
$0 < b < 1$	Exponential Decay	Radioactive decay
$b = 1$	No change (constant)	—

Growth Rate and Decay Rate

If value increases by $r\%$ per period: $f(x) = a(1 + r)^x$

If value decreases by $r\%$ per period: $f(x) = a(1 - r)^x$

Example: A car worth $25,000 depreciates by 15% per year: $V(t) = 25000(1 - 0.15)^t = 25000(0.85)^t$

Identifying Exponential Functions

Feature	Linear	Exponential
Pattern	Add a constant	Multiply by a constant
Equation	$y = mx + b$	$y = a \cdot b^x$
Rate of change	Constant	Changes (accelerating)
Graph	Straight line	Curved

From a Table

$x$	$y$ (Linear)	$y$ (Exponential)
0	5	5
1	8	10
2	11	20
3	14	40

Linear: $+3$ each time. Exponential: $\times 2$ each time.

Compound Interest

$A = P\left(1 + \frac{r}{n}\right)^{nt}$

Variable	Meaning
$A$	Final amount
$P$	Principal (initial)
$r$	Annual interest rate (decimal)
$n$	Times compounded per year
$t$	Time in years

Special case — continuous compounding: $A = Pe^{rt}$

Half-Life and Doubling Time

Half-life: $A = A_0 \left(\frac{1}{2}\right)^{t/h}$ where $h$ is the half-life

Doubling time: $A = A_0 \cdot 2^{t/d}$ where $d$ is the doubling time

SAT Question Types

Type 1: Interpret the Function

"In $f(t) = 500(1.03)^t$ , what does the 500 represent? What does 1.03 represent?"

500 = initial value
1.03 = 3% growth per period ( $b = 1 + r = 1 + 0.03$ )

Type 2: Growth/Decay Identification

"Does $y = 200(0.85)^x$ represent growth or decay, and by what percent?"

Decay (because $0.85 < 1$ )
Rate = $1 - 0.85 = 0.15 = 15\%$ decay per period

Type 3: Evaluate at a Specific Time

"If $P(t) = 1000(1.05)^t$ , what is $P(10)$ ?" $P(10) = 1000(1.05)^{10} \approx 1628.89$

Type 4: Compound Interest

"$5,000 at 4% annual interest compounded quarterly for 3 years" $A = 5000(1 + 0.04/4)^{4 \cdot 3} = 5000(1.01)^{12}$

Common SAT Mistakes

Confusing growth rate with growth factor: 5% growth has factor $1.05$ , not $0.05$
Forgetting the initial value: $f(0) = a$ , the initial value is the coefficient
Using the wrong formula for compounding: Check what $n$ is (monthly = 12, quarterly = 4, etc.)
Confusing linear and exponential — check if the table adds or multiplies
Not recognizing half-life pattern: Multiply by $\frac{1}{2}$ each half-life period

📚 Practice Problems

1Problem 1easy

❓ Question:

A bacteria colony starts with 100 bacteria and doubles every hour. Write a function for the population $P$ after $t$ hours, and find the population after 5 hours.

💡 Show Solution

Step 1: Identify the components:

Initial value: $a = 100$
Growth factor: $b = 2$ (doubling)

Function: $P(t) = 100 \cdot 2^t$

Step 2: Find $P(5)$ : $P(5) = 100 \cdot 2^5 = 100 \cdot 32 = 3{,}200$

Answer: $P(t) = 100 \cdot 2^t$ ; after 5 hours there are 3,200 bacteria.

2Problem 2easy

❓ Question:

A bacteria colony starts with 100 bacteria and doubles every hour. Write a function for the population $P$ after $t$ hours, and find the population after 5 hours.

💡 Show Solution

Step 1: Identify the components:

Initial value: $a = 100$
Growth factor: $b = 2$ (doubling)

Function: $P(t) = 100 \cdot 2^t$

Step 2: Find $P(5)$ : $P(5) = 100 \cdot 2^5 = 100 \cdot 32 = 3{,}200$

Answer: $P(t) = 100 \cdot 2^t$ ; after 5 hours there are 3,200 bacteria.

3Problem 3medium

❓ Question:

The value of a car is modeled by $V(t) = 30000(0.82)^t$ , where $t$ is in years. What is the annual depreciation rate, and what will the car be worth after 3 years?

💡 Show Solution

Step 1: Identify the decay rate. The base is $0.82$ , so: $r = 1 - 0.82 = 0.18 = 18\%$

The car depreciates by 18% per year.

Step 2: Find $V(3)$ : $V(3) = 30000(0.82)^3 = 30000(0.551368) \approx \$16{,}541$

Answer: 18% annual depreciation; worth approximately $16,541 after 3 years.

4Problem 4medium

❓ Question:

The value of a car is modeled by $V(t) = 30000(0.82)^t$ , where $t$ is in years. What is the annual depreciation rate, and what will the car be worth after 3 years?

💡 Show Solution

Step 1: Identify the decay rate. The base is $0.82$ , so: $r = 1 - 0.82 = 0.18 = 18\%$

The car depreciates by 18% per year.

Step 2: Find $V(3)$ : $V(3) = 30000(0.82)^3 = 30000(0.551368) \approx \$16{,}541$

Answer: 18% annual depreciation; worth approximately $16,541 after 3 years.

5Problem 5medium

❓ Question:

Which function represents a quantity that increases by 7% each month?

A) $f(x) = 100(7)^x$ B) $f(x) = 100(1.7)^x$ C) $f(x) = 100(0.07)^x$ D) $f(x) = 100(1.07)^x$

💡 Show Solution

Key: "Increases by 7% each month" means growth rate $r = 0.07$ .

The growth factor is $b = 1 + r = 1 + 0.07 = 1.07$ .

A) $b = 7$ → this is 600% growth, not 7% ✗ B) $b = 1.7$ → this is 70% growth, not 7% ✗ C) $b = 0.07$ → this is decay (and extreme decay at that) ✗ D) $b = 1.07$ → this is 7% growth ✓

Answer: D) $f(x) = 100(1.07)^x$

Common mistake: Using $0.07$ or $7$ as the base instead of $1.07$ .

6Problem 6medium

❓ Question:

Which function represents a quantity that increases by 7% each month?

A) $f(x) = 100(7)^x$ B) $f(x) = 100(1.7)^x$ C) $f(x) = 100(0.07)^x$ D) $f(x) = 100(1.07)^x$

💡 Show Solution

Key: "Increases by 7% each month" means growth rate $r = 0.07$ .

The growth factor is $b = 1 + r = 1 + 0.07 = 1.07$ .

A) $b = 7$ → this is 600% growth, not 7% ✗ B) $b = 1.7$ → this is 70% growth, not 7% ✗ C) $b = 0.07$ → this is decay (and extreme decay at that) ✗ D) $b = 1.07$ → this is 7% growth ✓

Answer: D) $f(x) = 100(1.07)^x$

Common mistake: Using $0.07$ or $7$ as the base instead of $1.07$ .

7Problem 7hard

❓ Question:

$2,000 is invested at 6% annual interest, compounded monthly. What is the value after 5 years?

💡 Show Solution

Formula: $A = P\left(1 + \frac{r}{n}\right)^{nt}$

Values:

$P = 2000$ (principal)
$r = 0.06$ (6% as decimal)
$n = 12$ (monthly compounding)
$t = 5$ (years)

Step 1: Substitute: $A = 2000\left(1 + \frac{0.06}{12}\right)^{12 \times 5}$ $A = 2000(1 + 0.005)^{60}$ $A = 2000(1.005)^{60}$

Step 2: Calculate: $(1.005)^{60} \approx 1.34885$ $A \approx 2000 \times 1.34885 \approx \$2{,}697.70$

Answer: Approximately $2,697.70

SAT Tip: Make sure to identify $n$ correctly: monthly = 12, quarterly = 4, semiannually = 2, annually = 1.

8Problem 8hard

❓ Question:

$2,000 is invested at 6% annual interest, compounded monthly. What is the value after 5 years?

💡 Show Solution

Formula: $A = P\left(1 + \frac{r}{n}\right)^{nt}$

Values:

$P = 2000$ (principal)
$r = 0.06$ (6% as decimal)
$n = 12$ (monthly compounding)
$t = 5$ (years)

Step 1: Substitute: $A = 2000\left(1 + \frac{0.06}{12}\right)^{12 \times 5}$ $A = 2000(1 + 0.005)^{60}$ $A = 2000(1.005)^{60}$

Step 2: Calculate: $(1.005)^{60} \approx 1.34885$ $A \approx 2000 \times 1.34885 \approx \$2{,}697.70$

Answer: Approximately $2,697.70

SAT Tip: Make sure to identify $n$ correctly: monthly = 12, quarterly = 4, semiannually = 2, annually = 1.

9Problem 9expert

❓ Question:

A radioactive substance has a half-life of 8 days. If you start with 500 grams, how much remains after 20 days?

💡 Show Solution

Half-life formula: $A = A_0 \left(\frac{1}{2}\right)^{t/h}$

Values:

$A_0 = 500$ grams
$h = 8$ days (half-life)
$t = 20$ days

Substitute: $A = 500\left(\frac{1}{2}\right)^{20/8} = 500\left(\frac{1}{2}\right)^{2.5}$

Calculate: $\left(\frac{1}{2}\right)^{2.5} = \left(\frac{1}{2}\right)^2 \cdot \left(\frac{1}{2}\right)^{0.5} = \frac{1}{4} \cdot \frac{1}{\sqrt{2}} = \frac{1}{4\sqrt{2}} \approx 0.17678$

$A \approx 500 \times 0.17678 \approx 88.4 \text{ grams}$

Answer: Approximately 88.4 grams

Quick check: After 8 days: 250g. After 16 days: 125g. After 24 days: 62.5g. So after 20 days (between 16 and 24), the answer should be between 62.5 and 125. ✓

10Problem 10expert

❓ Question:

A radioactive substance has a half-life of 8 days. If you start with 500 grams, how much remains after 20 days?

💡 Show Solution

Half-life formula: $A = A_0 \left(\frac{1}{2}\right)^{t/h}$

Values:

$A_0 = 500$ grams
$h = 8$ days (half-life)
$t = 20$ days

Substitute: $A = 500\left(\frac{1}{2}\right)^{20/8} = 500\left(\frac{1}{2}\right)^{2.5}$

Calculate: $\left(\frac{1}{2}\right)^{2.5} = \left(\frac{1}{2}\right)^2 \cdot \left(\frac{1}{2}\right)^{0.5} = \frac{1}{4} \cdot \frac{1}{\sqrt{2}} = \frac{1}{4\sqrt{2}} \approx 0.17678$

$A \approx 500 \times 0.17678 \approx 88.4 \text{ grams}$

Answer: Approximately 88.4 grams

Quick check: After 8 days: 250g. After 16 days: 125g. After 24 days: 62.5g. So after 20 days (between 16 and 24), the answer should be between 62.5 and 125. ✓

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❓ Frequently Asked Questions

What is Exponential Functions?▾

Master exponential growth and decay, interpret exponential expressions, solve exponential equations, and model real-world phenomena with exponential functions.

How can I study Exponential Functions effectively?▾

Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 10 problems provided, checking solutions as you go. Regular review and active practice are key to retention.

Is this Exponential Functions study guide free?▾

Yes — all study notes, flashcards, and practice problems for Exponential Functions on Study Mondo are free to access. No account is needed.

What course covers Exponential Functions?▾

Exponential Functions is part of the SAT Prep course on Study Mondo, specifically in the Advanced Math section. You can explore the full course for more related topics and practice resources.

Are there practice problems for Exponential Functions?▾

Yes, this page includes 10 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Exponential Functions

Try the Interactive Version!

Exponential Functions and Equations on the SAT

What Is an Exponential Function?

Growth vs. Decay

Growth Rate and Decay Rate

Identifying Exponential Functions

From a Table

Compound Interest

Half-Life and Doubling Time

SAT Question Types

Type 1: Interpret the Function

Type 2: Growth/Decay Identification

Type 3: Evaluate at a Specific Time

Type 4: Compound Interest

Common SAT Mistakes

📚 Practice Problems

1Problem 1easy

❓ Question:

2Problem 2easy

❓ Question:

3Problem 3medium

❓ Question:

4Problem 4medium

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5Problem 5medium

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6Problem 6medium

❓ Question:

7Problem 7hard

❓ Question:

8Problem 8hard

❓ Question:

9Problem 9expert

❓ Question:

10Problem 10expert

❓ Question:

Practice Lab

Practice with Flashcards

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❓ Frequently Asked Questions