Exponential Growth and Decay

📈 Exponential Growth and Decay

Part 1 of 5 — What Makes a Function Exponential?

Topics in This Part

🔑 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:

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

$y = 1 \cdot 2^{x}$

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

Concept Check 🎯

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

Every exponential function can be written as:

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

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

Example

$y = 5 \cdot 3^{x}$

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."

Identify $a$ and $b$ 🧮

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

1) $y = 7 \cdot 2^{x}$. $a = \,?$ 2) $y = 7 \cdot 2^{x}$. $b = \,?$ 3) $y = 100 \cdot (0.5)^{x}$. $a = \,?$

Exponential vs. Linear

Both grow, but they grow in completely different shapes.

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 $x^{2}$.

Linear or Exponential? 🔽

Classify each table or rule.

📈 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 decimal.

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

🔑 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:

$y = a(1 + r)^{x}$

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)

📈 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 subtract because you are losing a percent each step.

⚠️ 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:

$y = a(1 - r)^{x}$

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 🎯

📈 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: $y = 3 \cdot 2^{x}$

Notice the shape clues that show up on a graph:

💡 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)

📈 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

⚠️ 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.