🎯⭐ INTERACTIVE LESSON

Exponential Functions

Learn step-by-step with interactive practice!

← Back to Standard Lesson

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 🎯

Key Takeaways — Part 1

$f(t) = a \cdot b^t$ : $a$ = initial, $b$ = growth factor
Growth: (rate ). Decay: (rate )

Part 2: Exponential Decay

Exponential Functions

Part 2 of 7 — Compound Interest

The Compound Interest Formula

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

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

Part 4: Graphing Exponentials

Exponential Functions

Part 4 of 7 — Half-Life and Doubling Time

Half-Life

The amount remaining after time $t$ :

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

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	$y = mx + b$	$y = ab^x$

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: $P (t) = 100 {[(}^{}$

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

"Doubles every

k

" →

b^{t/k}

where

b = 2

Exponential eventually outpaces linear growth

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

$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)

Compound Interest 🎯

Key Takeaways — Part 2

Compound interest formula: $A = P(1 + r/n)^{nt}$
Identify $n$ from compounding frequency (annually, monthly, etc.)
Compound interest grows faster than simple interest over time
To find annual rate from the formula: $r = n \times (\text{base} - 1)$

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 🎯

Key Takeaways — Part 3

$y = b^x$ always passes through $(0, 1)$ with asymptote $y = 0$
Transformations shift the graph and asymptote: $y = a \cdot b^{x-h} + k$ has asymptote $y = k$
Growth ( $b > 1$ ): curve rises; Decay ( $0 < b < 1$ ): curve falls
Find the equation from two points: use $(0, y_0)$ for $a$ and another point for $b$

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

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

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 🎯

Key Takeaways — Part 4

Half-life: $A = A_0(1/2)^{t/h}$ . After $n$ half-lives, multiply by $(1/2)^n$
Doubling time: $A = A_0 \cdot 2^{t/d}$ . After $n$ doublings, multiply by $2^{}$
Count the number of half-lives or doublings first for quick mental math
Finding half-life from percent decay: solve $b^h = 0.5$

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

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

"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 🎯

Key Takeaways — Part 5

Linear: constant differences between consecutive y-values
Exponential: constant ratios between consecutive y-values
"Increases BY [amount]" → linear; "Increases BY [percent]" → exponential
Exponential always overtakes linear eventually

P (t) = 100 [(1.02)^{12}]^{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: $f(t) = 500(1.06)^{t} = 500((1.06)^{1/4})^{4t} \approx 500(1.01467)^{4t}$

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 🎯

Key Takeaways — Part 6

Rewrite $a \cdot b^{ct}$ as $a \cdot (b^c)^t$ to find the rate per unit time
Rewrite $a \cdot b^{t/c}$ as $a \cdot (b^{1/c})^t$ similarly
Monthly rate ≠ annual rate ÷ 12 (compounding makes it higher)
Exponent $nt$ : the base gives the rate per $1/n$ of a time unit

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 🎯

Key Takeaways — Part 7

The base determines the growth/decay rate — larger base = faster growth
Period: look at the exponent — $t/k$ means the rate applies over $k$ time units
To find $b$ from two points: $b = (y_2/y_1)^{1/(x_2-x_1)}$
For large values of $t$ , the base matters more than the initial value

Exponential Functions

Common Compounding Periods

Example

Continuous Compounding (Rare on SAT)

Key Takeaways — Part 2

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

Reading Exponential Graphs on SAT

Key Takeaways — Part 3

Doubling Time

Finding Half-Life from Decay Rate

Key Takeaways — Part 4

From a Table

SAT Question Type

The Key Difference for Word Problems

Key Takeaways — Part 5

Converting Between Growth Periods

Key Trick for SAT

Key Takeaways — Part 6

Interpreting in Context

Hard SAT Pattern: Finding the Equation from Context

Key Takeaways — Part 7

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 ⚠️

Key Takeaways — Part 1

Part 2: Exponential Decay

Exponential Functions

The Compound Interest Formula

Part 3: Compound Interest

Exponential Functions

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

Part 4: Graphing Exponentials

Exponential Functions

Half-Life

Part 5: Exponential Equations

Exponential Functions

How to Tell the Difference

Part 6: Problem-Solving Workshop

Exponential Functions

Changing the Base

Part 7: Review & Applications

Exponential Functions

Complete Exponential Toolkit

Common Compounding Periods

Example

Continuous Compounding (Rare on SAT)

Key Takeaways — Part 2

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

Reading Exponential Graphs on SAT

Key Takeaways — Part 3

Doubling Time

Finding Half-Life from Decay Rate

Key Takeaways — Part 4

From a Table

SAT Question Type

The Key Difference for Word Problems

Key Takeaways — Part 5

Converting Between Growth Periods

Key Trick for SAT

Key Takeaways — Part 6

Interpreting in Context

Hard SAT Pattern: Finding the Equation from Context

Key Takeaways — Part 7

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