🎯⭐ INTERACTIVE LESSON

Exponential Growth and Decay

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Exponential Growth and Decay - Complete Interactive Lesson

Part 1: What Makes a Function Exponential?

📈 Exponential Growth and Decay

Part 1 of 5 — What Makes a Function Exponential?

Topics in This Part

Section
Repeated Multiplication
The Form $y = a \cdot b^{x}$
Exponential vs. Linear

🔑 Key Concept: A function is exponential when each step multiplies by a fixed number instead of adding one. That single difference is what makes money, populations, and viruses explode — or fade away.

Repeated Multiplication

A linear pattern grows by adding the same amount each step. An exponential pattern grows by multiplying by the same amount each step.

Suppose a single bacterium doubles every hour:

Hour ( $x$ )	Bacteria ( $y$ )	How we got it
0	1	start
1	2	$1 \times 2$
2

Concept Check 🎯

The Form $y = a \cdot b^{x}$

Every exponential function can be written as:

$y = a \cdot b^{x}$

Identify $a$ and $b$ 🧮

For each function, read off the initial value $a$ and the base $b$ .

1) $y = 7 \cdot 2^{x}$ . . .

Exponential vs. Linear

Both grow, but they grow in completely different shapes.

	Linear $y = mx + b$	Exponential $y = a \cdot b^{x}$

Linear or Exponential? 🔽

Classify each table or rule.

Part 2: Exponential Growth

📈 Exponential Growth and Decay

Part 2 of 5 — Exponential Growth

🔑 The Idea: When the base $b$ is greater than 1, the function grows. The amount keeps getting bigger because you multiply by something larger than itself each step.

Growth Factor and Percent Rate

For growth we write the base as:

$b = 1 + r$

where $r$ is the growth rate written as a .

Part 3: Exponential Decay

📈 Exponential Growth and Decay

Part 3 of 5 — Exponential Decay

🔑 The Idea: When the base $b$ is between 0 and 1, the function decays. You keep only a fraction of the amount each step, so the total shrinks toward zero.

Decay Factor and Percent Rate

For decay we write the base as:

$b = 1 - r$

where $r$ is the decay rate as a decimal. You because you are a percent each step.

Part 4: Tables, Graphs & Word Problems

📈 Exponential Growth and Decay

Part 4 of 5 — Tables, Graphs & Word Problems

🔑 Big Payoff: Once you can build the equation, you can fill in tables, sketch the graph, and answer real questions about money, populations, and medicine.

Reading Tables and Graphs

Plug values of $x$ into $y = a \cdot b^{x}$ to build a table, then plot the points.

Example:

Part 5: Mixed Practice & Mastery Check

📈 Exponential Growth and Decay

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) recognize exponential patterns, (2) build growth models, (3) build decay models, and (4) evaluate and interpret them. Let's put it all together.

Quick Reference

Goal	Key move
General form	$y = a \cdot b^{x}$
Initial value	when (the -intercept)

Every row is the previous row times 2. Instead of writing all those multiplications, we use an exponent:

🔑 Key Idea: The exponent counts how many times you multiply. After $x$ hours you have multiplied by $2$ a total of $x$ times, so $y = 2^{x}$ .

Symbol	Name	What it means
$a$	initial value	the value of $y$ when $x = 0$ (the start)
$b$	base (growth/decay factor)	the number you multiply by each step
$x$	exponent	the number of steps (often time)

Why is $a$ the start? When $x = 0$ , $\;b^{0} = 1$ , so $y = a \cdot 1 = a$ .

Here $a = 5$ (you start at $5$ ) and $b = 3$ (you triple every step).

💡 Read it like a sentence: "Start at $a$ , and multiply by $b$ each time."

y = 100 \cdot (0.5)^{x}

Quick test: Look at a $y$ -table. Divide each value by the one before it.

If those ratios are all equal → exponential.
If the differences are all equal → linear.

⚠️ Common confusion: " $y = x^{2}$ " is not exponential — the variable is in the base there, not the exponent. Exponential means the variable is up in the exponent: $2^{x}$ , not $x^{2}$ .

The " $1$ " keeps the whole amount you already had; the " $+\,r$ " adds the new growth on top.

Percent increase	Rate $r$ (decimal)	Growth factor $b = 1 + r$
$5\%$	$0.05$	$1.05$
$20\%$	$0.20$	$1.20$
$50\%$	$0.50$	$1.50$
$100\%$	$1.00$	$2.00$ (doubles!)

🔑 Key Idea: A $7\%$ increase means you keep $100\%$ and add $7\%$ , so you multiply by $1.07$ each step — not by $0.07$ .

So the full growth model is:

Find the Growth Factor 🧮

Write the growth factor $b = 1 + r$ for each percent increase. (Enter a decimal.)

1) A town grows $3\%$ per year. $b = \,?$ 2) An investment grows $12\%$ per year. $b = \,?$ 3) A population grows $100\%$ per generation. $b = \,?$

Building a Growth Equation

To model growth, plug the start into $a$ and the factor into $b$ .

Example: A $2{,}000 deposit earns $4\%$ interest per year.

Initial value: $a = 2000$
Growth rate: $r = 0.04$ , so $b = 1.04$

$y = 2000(1.04)^{x}$

where $x$ is the number of years and $y$ is the balance.

Example: A video gets $50$ views, and views grow $20\%$ per day.

$a = 50$ , $\;r = 0.20$ , $\;b = 1.20$

$y = 50(1.20)^{x}$

💡 Pattern: $y = (\text{start}) \cdot (1 + \text{rate})^{\,\text{time}}$ . Get those three pieces and you have the equation.

Concept Check 🎯

Evaluate the Growth Model 🧮

Use $y = 2000(1.04)^{x}$ (the $2{,}000 deposit at $4\%$ ).

1) Balance after $0$ years: $y = \,?$ 2) Balance after $1$ year: $y = \,?$ 3) Balance after $2$ years: $y = \,?$ (round to the nearest cent if needed)

Percent decrease	Rate $r$ (decimal)	Decay factor $b = 1 - r$
$10\%$	$0.10$	$0.90$
$25\%$	$0.25$	$0.75$
$50\%$	$0.50$	$0.50$ (halves!)
$3\%$	$0.03$	$0.97$

⚠️ Watch out: A $10\%$ loss means you keep $90\%$ , so $b = 0.90$ , not $0.10$ . The factor is always what you keep, not what you lose.

The full decay model is:

Find the Decay Factor 🧮

Write the decay factor $b = 1 - r$ for each percent decrease. (Enter a decimal.)

1) A car loses $15\%$ of its value per year. $b = \,?$ 2) A medication leaves the body, dropping $30\%$ per hour. $b = \,?$ 3) A radioactive sample loses $50\%$ each period. $b = \,?$

Growth or Decay? 🔽

Decide from the base $b$ . Remember: $b > 1$ grows, $0 < b < 1$ decays.

Building a Decay Equation

Example: A $24{,}000 car loses $15\%$ of its value each year.

Initial value: $a = 24000$
Decay rate: $r = 0.15$ , so $b = 1 - 0.15 = 0.85$

$y = 24000(0.85)^{x}$

Example: A drug starts at $200$ mg and $25\%$ leaves the body each hour.

$a = 200$ , $\;r = 0.25$ , $\;b = 0.75$

$y = 200(0.75)^{x}$

💡 Pattern: $y = (\text{start}) \cdot (1 - \text{rate})^{\,\text{time}}$ . Same recipe as growth — just subtract the rate.

Evaluate the Decay Model 🧮

A drug starts at $200$ mg and $50\%$ leaves the body each hour, so $y = 200(0.5)^{x}$ .

1) Amount after $1$ hour: $y = \,?$ mg 2) Amount after $2$ hours: $y = \,?$ mg 3) Amount after $3$ hours: $y = \,?$ mg

Concept Check 🎯

$x$	$0$	$1$	$2$	$3$	$4$
$y$	$3$	$6$	$12$	$24$	$48$

Notice the shape clues that show up on a graph:

Feature	Growth ( $b > 1$ )	Decay ( $0 < b < 1$ )
$y$ -intercept	$(0, a)$	$(0, a)$
Direction	rises to the right	falls to the right
As $x$ gets large	shoots up	flattens toward $0$
Horizontal asymptote	$y = 0$ (to the left)	$y = 0$ (to the right)

💡 Both kinds always pass through $(0, a)$ — the initial value is the $y$ -intercept. And the curve never touches $y = 0$ ; it only gets infinitely close.

Fill the Table 🧮

Complete the table for $y = 3 \cdot 2^{x}$ .

1) $x = 0$ : $y = \,?$ 2) $x = 2$ : $y = \,?$ 3) $x = 4$ : $y = \,?$

Concept Check 🎯

Word Problems: A Full Walkthrough

Problem: A population of $400$ deer grows $5\%$ per year. How many after $3$ years?

Step 1 — Identify the pieces.

Start: $a = 400$
Rate: $r = 0.05$ (growth), so $b = 1.05$
Time: $x = 3$

Step 2 — Write the model. $y = 400(1.05)^{x}$

Step 3 — Substitute $x = 3$ . $y = 400(1.05)^{3} = 400(1.157625) = 463.05$

Step 4 — Interpret. About $463$ deer after $3$ years (round to a whole animal).

⚠️ Order of operations: Do the exponent first ( $1.05^{3}$ ), then multiply by $400$ . Never multiply $400 \times 1.05$ and then cube it.

Word-Problem Practice 🧮

A phone is worth $80 and loses $15\%$ of its value each year, so $y = 80(0.85)^{x}$ .

1) Value after $1$ year: $y = \,?$ dollars 2) Value after $2$ years: $y = \,?$ dollars (round to the nearest cent)

⚠️ Three traps to avoid:

A $7\%$ gain is $\times 1.07$ , not $\times 0.07$ .

A $7\%$ loss is $\times 0.93$ , not $\times 0.07$ .

$b^{0} = 1$ for any base, so the start is always $a$ .

Match the Model 🔽

Choose the correct base $b$ for each situation.

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.

Step rule	add a fixed amount	multiply by a fixed factor
First differences	constant	not constant
Ratios of values	not constant	constant
Graph shape	straight line	curve that bends upward (or hugs the axis)

Exponential Growth and Decay

Exponential Growth and Decay - Complete Interactive Lesson

Part 1: What Makes a Function Exponential?

📈 Exponential Growth and Decay

Topics in This Part

Repeated Multiplication

The Form y=a⋅bxy = a \cdot b^{x}y=a⋅bx

Exponential vs. Linear

Part 2: Exponential Growth

📈 Exponential Growth and Decay

Growth Factor and Percent Rate

Part 3: Exponential Decay

📈 Exponential Growth and Decay

Decay Factor and Percent Rate

Part 4: Tables, Graphs & Word Problems

📈 Exponential Growth and Decay

Reading Tables and Graphs

Example:

Part 5: Mixed Practice & Mastery Check

📈 Exponential Growth and Decay

Quick Reference

Example

Building a Growth Equation

Example: A $2{,}000 deposit earns 4%4\%4% interest per year.

Example: A video gets 505050 views, and views grow 20%20\%20% per day.

Building a Decay Equation

Example: A $24{,}000 car loses 15%15\%15% of its value each year.

Example: A drug starts at 200200200 mg and 25%25\%25% leaves the body each hour.

Word Problems: A Full Walkthrough

Problem: A population of 400400400 deer grows 5%5\%5% per year. How many after 333 years?

The Form $y = a \cdot b^{x}$

Example: A $2{,}000 deposit earns $4\%$ interest per year.

Example: A video gets $50$ views, and views grow $20\%$ per day.

Example: A $24{,}000 car loses $15\%$ of its value each year.

Example: A drug starts at $200$ mg and $25\%$ leaves the body each hour.

Problem: A population of $400$ deer grows $5\%$ per year. How many after $3$ years?