🎯⭐ INTERACTIVE LESSON

Exponential Functions

Learn step-by-step with interactive practice!

Exponential Functions - Complete Interactive Lesson

Part 1: Exponential Growth

Exponential Functions

Part 1 of 7 — Growth and Decay Models

Exponential Growth: $f(t) = a \cdot b^t$ where $b > 1$

$a$ = initial value (when $t = 0$ )
$b$ = growth factor
Growth rate: $r = b - 1$

Example: A population starts at 500 and grows 10% per year.

$P(t) = 500(1.10)^t$

Exponential Decay: $f(t) = a \cdot b^t$ where $0 < b < 1$

Decay rate: $r = 1 - b$

Example: A car worth $30,000 depreciates 15% per year.

$V(t) = 30000(0.85)^t$

Key Insight ⚠️

Exponential growth is NOT linear. It starts slow and gets dramatically fast.

Year	Linear (+100/yr)	Exponential (×1.5)
0	100	100
1	200	150
2	300	225
5	600	759
10	1,100	5,767

Growth & Decay 🎯

Worked Example 1 — Building the Model from a Story

A town's population is 8,000 and decreases by 3% per year. Write the model and find the population after 10 years.

Step	Work
Initial value	$a = 8000$
Decay rate	$3\% \Rightarrow b = 1 - 0.03 = 0.97$
Model	$P(t) = 8000(0.97)^t$
At $t = 10$	$8000(0.97)^{10} \approx 8000(0.7374) \approx 5{,}899$

Worked Example 2 — Period ≠ 1

A culture of 300 bacteria triples every 5 hours. How many after 20 hours?

Step	Work
Growth factor per period	$b = 3$ (triples)
Model	$N(t) = 300 \cdot 3^{t/5}$
At $t = 20$	$300 \cdot 3^{20/5} = 300 \cdot 3^4 = 300 \cdot 81 = 24{,}300$

Growth Factor Quick Reference

Phrasing	Growth Factor $b$
Increases by $20\%$	$1.20$
Decreases by $15\%$	$0.85$
Doubles	$2$
Triples	$3$
Loses half	$0.5$
Grows by a factor of 5	$5$

Growth & Decay Modeling 🎯

Identify the Model 🔍

For each scenario, pick the correct model type.

Key Takeaways — Part 1

Concept	Formula	Example
Growth ( $b > 1$ )	$f(t) = a \cdot b^t$	$P = 500(1.10)^t$
Decay ( $0 < b < 1$ )	$f(t) = a \cdot b^t$	$V = 30000(0.85)^t$
Growth rate from $b$	$r = b - 1$	$b = 1.10 \Rightarrow r = 10\%$
Decay rate from $b$	$r = 1 - b$	$b = 0.85 \Rightarrow r = 15\%$
Period ≠ 1	$a \cdot b^{t/k}$	Doubles every $k$ : $a \cdot 2^{t/k}$

"BY [amount]" → linear; "BY [percent]" → exponential
Exponential always eventually outpaces linear growth

Part 2: Exponential Decay

Exponential Functions

Part 2 of 7 — Compound Interest

The Compound Interest Formula

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

Variable	Meaning
$A$	Final amount
$P$	Principal (starting amount)
$r$	Annual interest rate (as decimal)
$n$	Number of times compounded per year
$t$	Number of years

Common Compounding Periods

$n$	Compounding
1	Annually
4	Quarterly
12	Monthly
365	Daily

Example

$5,000 invested at 6% compounded monthly for 3 years:

$A = 5000\left(1 + \frac{0.06}{12}\right)^{12 \times 3} = 5000(1.005)^{36} \approx \$5{,}983.40$

Continuous Compounding (Rare on SAT)

$A = Pe^{rt}$

Compound Interest 🎯

Worked Example 1 — Identifying Components

$A = 3000\left(1 + \frac{0.08}{4}\right)^{4 \times 5}$

Component	Value	Meaning
$P$	$3,000	Initial investment
$r$	$0.08$	8% annual rate
$n$	$4$	Compounded quarterly
$t$	$5$	5 years
$r/n$	$0.02$	2% per quarter
$nt$	$20$	20 total compounding periods

Worked Example 2 — Comparing Compounding Frequencies

$10,000 at 6% for 1 year:

Compounding	Calculation	Result
Annually ( $n=1$ )	$10000(1.06)^1$	$10,600.00
Monthly ( $n=12$ )	$10000(1.005)^{12}$	$10,616.78
Daily ( $n=365$ )	$10000(1.000164)^{365}$	$10,618.31

More frequent compounding → slightly more interest, but diminishing returns.

Simple vs. Compound Interest

	Simple	Compound
Formula	$A = P(1 + rt)$	$A = P(1 + r/n)^{nt}$
Growth	Linear	Exponential
Interest earned on	Principal only	Principal + prior interest

Interest Applications 🎯

Decode the Formula 🔍

Identify the meaning of each part.

Key Takeaways — Part 2

Concept	Formula / Rule
Compound interest	$A = P(1 + r/n)^{nt}$
Simple interest	$A = P(1 + rt)$
Find annual rate	$r = n \times (\text{period rate})$
Period rate	$r/n$ = base minus 1
More compounding	Slightly more interest (diminishing returns)

If you see...	It means...
$(1.02)^{12t}$	2% per month, compounded monthly
$(1.06)^t$	6% per year, compounded annually
$(1.015)^{4t}$	1.5% per quarter, compounded quarterly

To find the annual rate: multiply the period rate by $n$

Part 3: Compound Interest

Exponential Functions

Part 3 of 7 — Graphs of Exponential Functions

The Basic Graph: $y = b^x$

Growth ( $b > 1$ ): rises from left to right
Decay ( $0 < b < 1$ ): falls from left to right
Always passes through $(0, 1)$ since $b^0 = 1$
Horizontal asymptote: $y = 0$ (the x-axis)
Domain: all real numbers; Range: $y > 0$

Transformations: $y = a \cdot b^{x-h} + k$

Parameter	Effect
$a$	Vertical stretch/flip (if negative: reflected)
$h$	Horizontal shift (right if positive)
$k$	Vertical shift (up if positive)
$k$	New horizontal asymptote: $y = k$

Reading Exponential Graphs on SAT

From a graph, identify:

y-intercept: the initial value $a$ (where the graph crosses $y$ -axis)
Horizontal asymptote: the value $y$ approaches but never reaches
Growth vs decay: is the function increasing or decreasing?
Growth factor: pick two integer $x$ -values, divide $y$ -values

Exponential Graphs 🎯

Worked Example 1 — Finding the Equation from a Graph

An exponential graph passes through $(0, 5)$ and $(2, 45)$ . Its asymptote is $y = 0$ . Find the equation.

Step	Work
At $x = 0$	$a \cdot b^0 = a = 5$
At $x = 2$	$5b^2 = 45 \Rightarrow b^2 = 9 \Rightarrow b = 3$
Answer	$y = 5(3)^x$

Worked Example 2 — Asymptote Shifted

An exponential graph has asymptote $y = 2$ , y-intercept at $(0, 5)$ , and passes through $(1, 8)$ .

Step	Work
Form	$y = a \cdot b^x + k$ , where $k = 2$
At $x = 0$	$a + 2 = 5 \Rightarrow a = 3$
At $x = 1$	$3b + 2 = 8 \Rightarrow 3b = 6 \Rightarrow b = 2$
Answer	$y = 3(2)^x + 2$

Exponential vs. Other Graphs

Feature	Exponential	Linear	Quadratic
Shape	J-curve or reverse-J	Straight line	Parabola (U or ∩)
Asymptote	Yes (horizontal)	No	No
Passes through $(0,1)$	$y = b^x$ only	Only if $b = 1$	Only if $c = 1$
End behavior	One end → ∞, other → asymptote	Both ends → ±∞	Both ends → ∞ or → −∞

Reading Exponential Graphs 🎯

Graph Feature Analysis 🔍

Identify each graph feature for the given function.

Key Takeaways — Part 3

Feature	$y = a \cdot b^{x-h} + k$
y-intercept	Plug $x = 0$ : $a \cdot b^{-h} + k$
Asymptote	$y = k$
Growth vs. decay	$b > 1$ : growth; $0 < b < 1$ : decay
Domain	All real numbers
Range	$y > k$ (if $a > 0$ ) or $y < k$ (if $a < 0$ )

Finding equation from graph
1. Read asymptote → gives $k$
2. Read y-intercept → solve for $a$
3. Use another point → solve for $b$

Part 4: Graphing Exponentials

Exponential Functions

Part 4 of 7 — Half-Life and Doubling Time

Half-Life

The amount remaining after time $t$ :

$A(t) = A_0 \left(\frac{1}{2}\right)^{t/h}$

Where $h$ = half-life (time to lose half).

Example: A 400g sample has a half-life of 5 days.

After 15 days: $A = 400(1/2)^{15/5} = 400(1/2)^3 = 400(1/8) = 50$ grams

Doubling Time

$A(t) = A_0 \cdot 2^{t/d}$

Where $d$ = doubling time.

Example: A population of 1000 doubles every 7 years.

After 21 years: $A = 1000 \cdot 2^{21/7} = 1000 \cdot 2^3 = 8000$

Finding Half-Life from Decay Rate

If something decays by $r\%$ per period:

Decay factor: $b = 1 - r/100$
Half-life: solve $b^h = 1/2$ → $h = \frac{\ln(1/2)}{\ln(b)}$

On the SAT, you can often solve by testing: "After how many periods does the amount drop below half?"

Half-Life & Doubling 🎯

Worked Example 1 — Counting Half-Lives

1,200 grams with a half-life of 8 hours. How much after 1 day (24 hours)?

Step	Work
Number of half-lives	$24 / 8 = 3$
After 1 half-life	$1200 / 2 = 600$
After 2 half-lives	$600 / 2 = 300$
After 3 half-lives	$300 / 2 = 150$

Or: $1200 \times (1/2)^3 = 1200 / 8 = 150$ grams.

Worked Example 2 — Finding Doubling Time

A population grows 12% per year. How long until it doubles?

Step	Work
Model	$(1.12)^t = 2$
Test $t = 6$	$(1.12)^6 \approx 1.974$ (just under 2)
Test $t = 7$	$(1.12)^7 \approx 2.211$ (just over 2)
Answer	About 6 years

Rule of 70: Doubling time $\approx 70 / (\text{percent rate})$ . For $12\%$ : $70/12 \approx 5.8$ years ✓

Half-Life Table

Half-Lives	Fraction Remaining	Decimal
0	$1$	$1.000$
1	$1/2$	$0.500$
2	$1/4$	$0.250$
3	$1/8$	$0.125$
4	$1/16$	$0.0625$
5	$1/32$	$0.03125$

Half-Life & Doubling Mastery 🎯

Half-Life Quick Calculations 🔍

How much remains?

Key Takeaways — Part 4

Concept	Formula	Quick Method
Half-life	$A = A_0(1/2)^{t/h}$	Count half-lives, divide by $2^n$
Doubling time	$A = A_0 \cdot 2^{t/d}$	Count doublings, multiply by $2^n$
Rule of 70	Doubling time $\approx 70/r\%$	For growth rate $r\%$ per period
Rule of 70 (reverse)	Rate $\approx 70/\text{doubling time}$	If doubling time is known

After $n$ half-lives: multiply by $(1/2)^n$
After $n$ doublings: multiply by $2^n$
Rule of 70 gives quick estimates without a calculator

Part 5: Exponential Equations

Exponential Functions

Part 5 of 7 — Exponential vs. Linear

How to Tell the Difference

Feature	Linear	Exponential
Pattern	Add constant	Multiply by constant
Formula	$y = mx + b$	$y = ab^x$
Table	Constant differences	Constant ratios
Graph	Straight line	Curve

From a Table

$x$	$y$ (linear)	$y$ (exponential)
0	3	3
1	7	6
2	11	12
3	15	24

Linear: differences are all $+4$ . Exponential: ratios are all $\times 2$ .

SAT Question Type

"Which type of function best models the data?"

Check: are the differences constant (linear) or are the ratios constant (exponential)?

The Key Difference for Word Problems

"Increases by 50 each year" → linear: $y = 50t + b$
"Increases by 50% each year" → exponential: $y = a(1.5)^t$

Linear vs. Exponential 🎯

Worked Example 1 — Table Analysis

Determine if this data is linear, exponential, or neither:

$x$	$y$	Difference	Ratio
0	4	—	—
1	12	$+8$	$3$
2	36	$+24$	$3$
3	108	$+72$	$3$

Differences are NOT constant → not linear. Ratios ARE constant ( $\times 3$ ) → exponential: $y = 4(3)^x$ .

Worked Example 2 — SAT Word Problem

"A tank starts with 200 gallons and loses 25 gallons per hour." Linear or exponential?

Clue	Interpretation
"Loses 25 gallons"	Fixed amount → linear
Model	$y = 200 - 25t$

Compare: "A tank starts with 200 gallons and loses 25% per hour" → exponential: $y = 200(0.75)^t$ .

When They Cross

Linear and exponential functions may be equal at certain points, but exponential always wins for large $t$ :

$t$	$L(t) = 100 + 50t$	$E(t) = 10(1.5)^t$
0	100	10
5	350	75.9
10	600	576.7
15	850	4,379
20	1,100	33,252

Model Identification 🎯

Linear or Exponential? 🔍

Classify each scenario.

Key Takeaways — Part 5

Feature	Linear	Exponential
Key word	"by [amount]"	"by [percent]" or "times"
Table test	Constant differences	Constant ratios
Formula	$y = mx + b$	$y = ab^x$
Graph	Straight line	Curve with asymptote
Long-term	Steady growth	Explosive growth

For SAT data tables: check ratios first (divide consecutive $y$ -values)
If ratios are constant → exponential; if differences are constant → linear
Exponential ALWAYS beats linear for large enough $x$

Part 6: Problem-Solving Workshop

Exponential Functions

Part 6 of 7 — Rewriting Exponential Expressions

Changing the Base

The SAT often asks you to rewrite exponentials in equivalent forms.

Example: Express the annual growth rate from a monthly model:

$P(t) = 100(1.02)^{12t}$

Rewrite: $P(t) = 100 \left[(1.02)^{12}\right]^t = 100(1.2682)^t$

So the monthly rate is 2% but the annual rate is about 26.82%.

Converting Between Growth Periods

$f(t) = 500(1.06)^t$ (annual growth of 6%)

Quarterly equivalent: $f(t) = 500(1.06)^{t} = 500((1.06)^{1/4})^{4t} \approx 500(1.01467)^{4t}$

Key Trick for SAT

If you see $(1.03)^{4t}$ :

This means 3% growth per quarter (since the exponent is $4t$ )
Annual rate: $(1.03)^4 - 1 \approx 12.55\%$

If you see $(0.95)^{t/2}$ :

This means 5% decay every 2 years (since the exponent is $t/2$ )
Annual rate: $(0.95)^{1/2} - 1 \approx -2.53\%$

Rewriting Exponentials 🎯

Worked Example 1 — What Does the Base Represent?

$P(t) = 2000(1.03)^{4t}$ . What does $1.03$ represent?

Analysis	Meaning
Exponent $4t$	4 compounding periods per year → quarterly
Base $1.03$	Growth factor per quarter
Interpretation	"The quantity grows by 3% each quarter"

To find annual rate: Rewrite as $2000[(1.03)^4]^t = 2000(1.1255)^t$ → annual rate ≈ $12.55\%$ .

Worked Example 2 — Rewriting for a Different Period

$A(t) = 500(2)^{t/10}$ models doubling every 10 years. What is the yearly growth factor?

Step	Work
Rewrite	$500(2^{1/10})^t$
Calculate $2^{1/10}$	$\approx 1.0718$
Interpretation	About $7.18\%$ growth per year

Common SAT Rewrite Patterns

Given Form	Rewritten as $a \cdot c^t$	Period Rate
$a(1.02)^{12t}$	$a(1.2682)^t$	Monthly rate 2%
$a(0.95)^{t/3}$	$a(0.9830)^t$	Decays ~5% every 3 periods
$a(3)^{t/5}$	$a(1.2457)^t$	Triples every 5 periods

Equivalence & Interpretation 🎯

Interpret the Exponent 🔍

What does the exponent structure tell you about the time period?

Key Takeaways — Part 6

Rewrite Goal	Method
Find rate per unit time	$a \cdot b^{ct} = a(b^c)^t$ ; rate $= b^c - 1$
Find rate per longer period	$a \cdot b^{t/c} = a(b^{1/c})^t$ ; rate $= b^{1/c} - 1$
Convert monthly → annual	$(1 + r_{\text{monthly}})^{12} - 1$
Convert annual → monthly	$(1 + r_{\text{annual}})^{1/12} - 1$

SAT Interpretation Pattern
$(1.03)^{4t}$ : "3% growth per quarter"
$(0.95)^{t/2}$ : "5% decay every 2 periods"
$(2)^{t/10}$ : "doubles every 10 periods"

Monthly rate × 12 ≠ annual rate (compounding makes it higher)
Always rewrite so the exponent is just $t$ to find the per-unit rate

Part 7: Review & Applications

Exponential Functions

Part 7 of 7 — Review & Hard Practice

Complete Exponential Toolkit

Model	Formula	Key Feature
Basic growth	$y = ab^t$ , $b > 1$	Constant percent increase
Basic decay	$y = ab^t$ , $0 < b < 1$	Constant percent decrease
Compound interest	$A = P(1 + r/n)^{nt}$	Interest on interest
Half-life	$A = A_0(1/2)^{t/h}$	Amount halves every $h$
Doubling	$A = A_0 \cdot 2^{t/d}$	Amount doubles every $d$

Interpreting in Context

When the SAT gives you $f(t) = 300(0.85)^{t/4}$ and asks what 0.85 means:

"The quantity decreases by 15% every 4 units of time."

The base tells you the rate; the denominator in the exponent tells you the period.

Hard SAT Pattern: Finding the Equation from Context

"A sample decreases from 200 to 50 in 6 hours."

$50 = 200 \cdot b^6$ → $b^6 = 1/4$ → $b = (1/4)^{1/6} = 4^{-1/6} = 2^{-1/3}$

Or: $b^6 = 0.25$ → $b = 0.25^{1/6} \approx 0.794$

So every hour, about 20.6% decays.

Mixed Review 🎯

Worked Example 1 — Finding $b$ from Two Points

An exponential function passes through $(2, 18)$ and $(5, 486)$ . Find the equation.

Step	Work
Use ratio	$\frac{f(5)}{f(2)} = \frac{ab^5}{ab^2} = b^3 = \frac{486}{18} = 27$
Solve for $b$	$b = 27^{1/3} = 3$
Solve for $a$	$a \cdot 3^2 = 18 \Rightarrow 9a = 18 \Rightarrow a = 2$
Answer	$f(t) = 2(3)^t$

Worked Example 2 — Percent Change from a Model

$f(t) = 800(0.92)^t$ — What does this model?

Reading	Meaning
$a = 800$	Starts at 800
$b = 0.92$	Retains 92% each period → loses 8%
At $t = 5$	$800(0.92)^5 \approx 800(0.6591) \approx 527$
After 5 periods	About 34% has been lost

SAT Exponential Strategy Summary

Question Type	Strategy
"What is the initial value?"	Find $a$ (coefficient)
"What is the growth/decay rate?"	Rate $=
"What does the base represent?"	Check exponent structure for period
"Which model is exponential?"	Look for constant ratios or percent change
"Rewrite in different form"	Use exponent rules: $b^{ct} = (b^c)^t$
"When does it reach [value]?"	Solve $ab^t = \text{value}$ , test values

SAT Challenge Round 🎯

Exponential Mastery Check 🔍

Answer each quick question.

Key Takeaways — Part 7

Part	Core Skill
1	Growth/decay models: $a \cdot b^t$ , identifying rate from base
2	Compound interest: $P(1 + r/n)^{nt}$ , reading period rates
3	Graphs: asymptotes, y-intercepts, finding equation from graph
4	Half-life & doubling: counting periods, Rule of 70
5	Linear vs. exponential: differences vs. ratios
6	Rewriting: converting between time periods
7	Review: finding $b$ from two points, SAT strategy

Exponential Quick-Reference

Formula	Used For
$y = ab^t$	General growth/decay
$A = P(1 + r/n)^{nt}$	Compound interest
$A = A_0(1/2)^{t/h}$	Half-life
$A = A_0 \cdot 2^{t/d}$	Doubling
Rule of 70: $d \approx 70/r\%$	Estimate doubling time
$b = (y_2/y_1)^{1/(x_2 - x_1)}$	Find base from two points

← Back to Standard Lesson

Exponential Functions

Exponential Functions - Complete Interactive Lesson

Part 1: Exponential Growth

Exponential Functions

Exponential Growth: f(t)=a⋅btf(t) = a \cdot b^tf(t)=a⋅bt where b>1b > 1b>1

Exponential Decay: f(t)=a⋅btf(t) = a \cdot b^tf(t)=a⋅bt where 0<b<10 < b < 10<b<1

Key Insight ⚠️

Worked Example 1 — Building the Model from a Story

Worked Example 2 — Period ≠ 1

Growth Factor Quick Reference

Key Takeaways — Part 1

Part 2: Exponential Decay

Exponential Functions

The Compound Interest Formula

Common Compounding Periods

Example

Continuous Compounding (Rare on SAT)

Worked Example 1 — Identifying Components

Worked Example 2 — Comparing Compounding Frequencies

Simple vs. Compound Interest

Key Takeaways — Part 2

Part 3: Compound Interest

Exponential Functions

The Basic Graph: y=bxy = b^xy=bx

Transformations: y=a⋅bx−h+ky = a \cdot b^{x-h} + ky=a⋅bx−h+k

Reading Exponential Graphs on SAT

Worked Example 1 — Finding the Equation from a Graph

Worked Example 2 — Asymptote Shifted

Exponential vs. Other Graphs

Key Takeaways — Part 3

Part 4: Graphing Exponentials

Exponential Functions

Half-Life

Doubling Time

Finding Half-Life from Decay Rate

Worked Example 1 — Counting Half-Lives

Worked Example 2 — Finding Doubling Time

Half-Life Table

Key Takeaways — Part 4

Part 5: Exponential Equations

Exponential Functions

How to Tell the Difference

From a Table

SAT Question Type

The Key Difference for Word Problems

Worked Example 1 — Table Analysis

Worked Example 2 — SAT Word Problem

When They Cross

Key Takeaways — Part 5

Part 6: Problem-Solving Workshop

Exponential Functions

Changing the Base

Converting Between Growth Periods

Key Trick for SAT

Worked Example 1 — What Does the Base Represent?

Worked Example 2 — Rewriting for a Different Period

Common SAT Rewrite Patterns

Key Takeaways — Part 6

Part 7: Review & Applications

Exponential Functions

Complete Exponential Toolkit

Interpreting in Context

Hard SAT Pattern: Finding the Equation from Context

Worked Example 1 — Finding bbb from Two Points

Worked Example 2 — Percent Change from a Model

SAT Exponential Strategy Summary

Key Takeaways — Part 7

Exponential Quick-Reference

Exponential Functions

Exponential Functions - Complete Interactive Lesson

Part 1: Exponential Growth

Exponential Functions

Exponential Growth: f(t)=a⋅btf(t) = a \cdot b^tf(t)=a⋅bt where b>1b > 1b>1

Exponential Decay: f(t)=a⋅btf(t) = a \cdot b^tf(t)=a⋅bt where 0<b<10 < b < 10<b<1

Key Insight ⚠️

Worked Example 1 — Building the Model from a Story

Worked Example 2 — Period ≠ 1

Growth Factor Quick Reference

Key Takeaways — Part 1

Part 2: Exponential Decay

Exponential Growth: $f(t) = a \cdot b^t$ where $b > 1$

Exponential Decay: $f(t) = a \cdot b^t$ where $0 < b < 1$

The Basic Graph: $y = b^x$

Transformations: $y = a \cdot b^{x-h} + k$

Worked Example 1 — Finding $b$ from Two Points

Exponential Growth: $f(t) = a \cdot b^t$ where $b > 1$

Exponential Decay: $f(t) = a \cdot b^t$ where $0 < b < 1$

The Basic Graph: $y = b^x$

Transformations: $y = a \cdot b^{x-h} + k$

Worked Example 1 — Finding $b$ from Two Points