Exponential Functions

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.

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.

Worked Example 2 — Period ≠ 1

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

Growth Factor Quick Reference

Growth & Decay Modeling 🎯

Identify the Model 🔍

For each scenario, pick the correct model type.

Key Takeaways — Part 1

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

Exponential Functions

Part 2 of 7 — Compound Interest

The Compound Interest Formula

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

Common Compounding Periods

Example

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

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

Reading the Exponent

The SAT often hides the compounding schedule in the exponent: $(1.005)^{12t}$ means a $0.5\%$ rate applied 12 times per year. Continuous compounding (with the constant $e$) is not tested on the digital SAT.

Compound Interest 🎯

Worked Example 1 — Identifying Components

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

Worked Example 2 — Comparing Compounding Frequencies

$10,000 at 6% for 1 year:

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

Simple vs. Compound Interest

Interest Applications 🎯

Decode the Formula 🔍

Identify the meaning of each part.

Key Takeaways — Part 2

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

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

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.

Worked Example 2 — Asymptote Shifted

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

Exponential vs. Other Graphs

Reading Exponential Graphs 🎯

Graph Feature Analysis 🔍

Identify each graph feature for the given function.

Key Takeaways — Part 3

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

Reading Half-Life from the Exponent

The SAT usually hands you the half-life inside the model. In $A = 500\left(\frac{1}{2}\right)^{t/6}$, the amount is multiplied by $\frac{1}{2}$ each time $t$ grows by $6$, so the half-life is $6$ units of time. In the same way, $A = 500 \cdot 2^{t/4}$ doubles every $4$ units.

Estimating Half-Life from a Percent Decay Rate

If something decays by $r\%$ per period, its decay factor is $b = 1 - \frac{r}{100}$. Multiply by $b$ one period at a time until the running product reaches about $0.5$.

Example: A sample loses $30\%$ per hour, so $b = 0.7$.

• After 1 hour: $0.7$ of the sample remains
• After 2 hours: $0.7^2 = 0.49$ remains, just under half

So the half-life is a little under 2 hours. No special formula is needed, just repeated multiplication.

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)?

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?

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

Half-Life Table

Half-Life & Doubling Mastery 🎯

Half-Life Quick Calculations 🔍

How much remains?

Key Takeaways — Part 4

• 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

Exponential Functions

Part 5 of 7 — Exponential vs. Linear

How to Tell the Difference

From a Table

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:

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?

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 $t$:

Model Identification 🎯

Linear or Exponential? 🔍

Classify each scenario.

Key Takeaways — Part 5

• 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 $x$

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?

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?

Common SAT Rewrite Patterns

Equivalence & Interpretation 🎯

Interpret the Exponent 🔍

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

Key Takeaways — Part 6

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

Exponential Functions

Part 7 of 7 — Review & Hard Practice

Complete Exponential Toolkit

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.

Worked Example 2 — Percent Change from a Model

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

SAT Exponential Strategy Summary

SAT Challenge Round 🎯

Exponential Mastery Check 🔍

Answer each quick question.

Key Takeaways — Part 7

Exponential Quick-Reference