Grade 9

Intermediate

7 min

Lesson 8 of 12

Exponential Growth and Decay

Not Started

Step into the world of rapid change where values double, triple, or shrink by a fixed percentage over time.

If you started with a single penny and doubled your money every day, would you rather have that penny or a flat $1,000,000 after 30 days? The answer might shock you—and it's all thanks to the explosive nature of exponential growth.

Learning Objectives

1.Distinguish between linear growth (adding) and exponential growth (multiplying) in data sets.
2.Identify the initial value $a$ and the growth/decay factor $b$ in the equation $y = a(b)^x$ .
3.Predict and sketch the general shape of exponential growth and decay curves.

Linear vs. Exponential: The Great Divide

In the world of math, change happens in two primary ways. Linear growth occurs when you add the same amount over and over. Think of a piggy bank where you add $5 every week. **Exponential growth**, however, occurs when you multiply by the same factor over and over. Instead of adding$ 5, imagine your money doubles every week. While linear growth looks like a steady climb, exponential growth starts slow and then 'explodes' upward. To spot the difference in a table, look at the $y$ -values: if you are adding, it's linear; if you are multiplying, it's exponential.

Quick Check

If a population of bacteria is 100, 200, 400, 800... is this linear or exponential?

Check if you are adding the same number or multiplying by the same number to get to the next step.

Answer

It is exponential because each value is being multiplied by 2, not increased by a constant sum.

Anatomy of the Exponential Equation

All exponential functions follow a specific blueprint:

y = a(b)^x

. Each part of this equation tells a story. The variable **

a

represents the initial value** (the starting point when

x = 0

). The variable **

b

is the growth or decay factor**. This is the number we multiply by at every step. If

b > 1

, the values are getting larger (growth). If

0 < b < 1

(a fraction or decimal like

0.5

), the values are shrinking (decay). Note that

b

can never be negative or zero in these models!

Given the equation $y = 500(1.05)^x$ , identify the starting value and the growth factor.

1. Look for the value of $a$ , which is the number outside the parentheses: $a = 500$ . This is our starting point. 2. Look for the value of $b$ , the base of the exponent: $b = 1.05$ . 3. Since $1.05 > 1$ , this represents a growth of $5\%$ per time period.

Quick Check

In the equation $y = 10(0.85)^x$ , is the value growing or decaying?

Look at the value inside the parentheses.

Answer

Decaying, because the factor $b$ (0.85) is less than 1.

Visualizing the Curve

Graphs of exponential functions have a distinct look. An exponential growth curve starts near the x-axis on the left and sweeps upward rapidly to the right, forming a 'J' shape. An exponential decay curve starts high on the left and drops quickly, flattening out as it approaches the x-axis on the right. A key feature is the asymptote: a line that the graph gets closer and closer to but never actually touches. For basic equations, this is usually the x-axis ( $y = 0$ ).

A new car costs $30,000 and loses$ 15\%$ of its value every year. Write the equation.

1. Identify the initial value $a$ : $a = 30,000$ . 2. Determine the decay factor $b$ . If it loses $15\%$ , it keeps $85\%$ of its value. So, $b = 1 - 0.15 = 0.85$ . 3. Plug into the formula: $y = 30,000(0.85)^x$ . 4. The graph would start at $(0, 30000)$ and curve downward toward the x-axis.

A radioactive substance has a half-life of 10 years. If you start with 80 grams, how much is left after 30 years?

1. Initial value $a = 80$ . 2. The factor $b = 0.5$ (because it is a 'half'-life). 3. The exponent $x$ represents the number of half-life periods. Since 30 years is three 10-year periods, $x = 3$ . 4. Equation: $y = 80(0.5)^3$ . 5. Solve: $y = 80(0.125) = 10$ grams.

Key Takeaways

1Linear functions add a constant; exponential functions multiply by a constant factor.
2The standard form is $y = a(b)^x$ , where $a$ is the start and $b$ is the multiplier.
3If $b > 1$ , the graph shows growth; if $0 < b < 1$ , the graph shows decay.
4Exponential graphs never touch their asymptote (usually the x-axis) because multiplying by a positive number never results in zero.

Quick Check

Multiple Choice

Which of these tables represents exponential growth?

Multiple Choice

In the function $y = 2(3)^x$ , what is the y-intercept?

True/False

The graph of $y = 100(0.5)^x$ will eventually cross below the x-axis and become negative.

What's Next?

Review Tomorrow

In 24 hours, try to write down the general formula for an exponential function and explain what 'a' and 'b' stand for without looking at your notes.

Practice Activity

Find a receipt or a news article mentioning inflation or interest rates. Try to determine if the change described is linear or exponential.

Browse

Browse

Exponential Growth and Decay

Learning Objectives

Linear vs. Exponential: The Great Divide

Anatomy of the Exponential Equation

Visualizing the Curve

Key Takeaways

Quick Check

What's Next?

Exponential Growth and Decay

Learning Objectives

Linear vs. Exponential: The Great Divide

Anatomy of the Exponential Equation

Visualizing the Curve

Key Takeaways

Quick Check

What's Next?