Grade 11

Advanced

7 min

Lesson 3 of 12

Exponential Functions and Growth

Not Started

Analyze exponential growth and decay models used in finance, biology, and physics.

Would you rather have $1,000,000 right now, or a single penny that doubles in value every day for 30 days? By the end of this lesson, you'll see why the penny is worth over$ 5 million.

Learning Objectives

1.Identify and graph exponential functions while locating their horizontal asymptotes.
2.Model real-world growth and decay using the formula $y = a(1 \pm r)^t$ .
3.Apply the natural base $e$ to solve continuous compounding problems in finance and science.

The Anatomy of Exponential Functions

Unlike linear functions that grow by adding a constant, exponential functions grow by multiplying by a constant factor. The general form is $f(x) = ab^x$ , where $a$ is the initial value (y-intercept) and $b$ is the base or growth factor. If $b > 1$ , the function represents exponential growth. If $0 < b < 1$ , it represents exponential decay. A critical feature of these graphs is the horizontal asymptote, usually at $y = 0$ , which the curve approaches but never touches as $x$ moves toward infinity or negative infinity.

Identify the initial value, growth factor, and asymptote for $f(x) = 3(2)^x$ .

1. The initial value $a$ is the coefficient, so $a = 3$ . 2. The base $b$ is $2$ . Since $2 > 1$ , this is a growth function. 3. There is no vertical shift ( $k=0$ ), so the horizontal asymptote is $y = 0$ .

Quick Check

If a function is defined as $f(x) = 5(0.75)^x$ , is it growth or decay, and what is the horizontal asymptote?

Look at the value of the base

b

Answer

It is exponential decay (because $0.75 < 1$ ) and the horizontal asymptote is $y = 0$ .

Modeling Real-World Change

In finance and biology, we often express the base

b

in terms of a rate of change

r

. For growth,

b = (1 + r)

; for decay,

b = (1 - r)

. This gives us the formula

y = a(1 \pm r)^t

. For example, if a population grows by

5\%

r = 0.05

and the multiplier is

1.05

. If a car depreciates by

15\%

r = 0.15

and the multiplier is

0.85

. This multiplicative scaling is why exponential growth starts slow but eventually explodes.

A car is purchased for $25,000 and loses$ 12\%$ of its value each year. How much is it worth after 5 years?

1. Identify variables: $a = 25000$ , $r = 0.12$ , $t = 5$ . 2. Set up the decay formula: $V(t) = 25000(1 - 0.12)^t$ . 3. Simplify: $V(5) = 25000(0.88)^5$ . 4. Calculate: $V(5) \approx 25000(0.5277) \approx$ 13,193.30.

Quick Check

If an investment grows by $8\%$ annually, what is the base $b$ used in the formula $ab^x$ ?

Use

1 + r

where

r

is the decimal form of

8\%

Answer

The base $b$ is $1.08$ .

The Natural Base $e$ and Continuous Growth

In nature, growth doesn't usually happen in discrete yearly jumps; it happens constantly. This brings us to the natural base

e \approx 2.71828

. The formula for continuous compounding is

A = Pe^{rt}

, where

P

is the principal,

r

is the rate, and

t

is time. This constant

e

is the limit of

(1 + \frac{1}{n})^n

n

approaches infinity. It is the 'speed limit' of growth and is used in everything from radioactive decay to calculating interest in high-frequency trading.

You invest $5,000 in an account paying$ 6\%$ interest compounded continuously. How much will you have after 10 years compared to simple annual compounding?

1. Continuous: $A = 5000e^{(0.06)(10)} = 5000e^{0.6} \approx 5000(1.8221) =$ 9,110.50. 2. Annual: $A = 5000(1 + 0.06)^{10} = 5000(1.06)^{10} \approx 5000(1.7908) =$ 8,954.24. 3. The continuous compounding yields $156.26 more due to the constant reinvestment of interest.

Key Takeaways

1Exponential functions follow the form $y = ab^x$ , where $b$ determines growth ( $b>1$ ) or decay ( $0<b<1$ ).
2The horizontal asymptote for $f(x) = ab^x + k$ is always the line $y = k$ .
3Continuous growth or decay is modeled using the natural base $e$ in the formula $A = Pe^{rt}$ .
4Linear growth is additive ( $+m$ ), while exponential growth is multiplicative ( $\times b$ ).

Quick Check

Multiple Choice

Which function represents exponential decay?

Multiple Choice

What is the horizontal asymptote of $f(x) = 4(3)^x - 5$ ?

True/False

Continuous compounding using $e$ will always result in a higher return than annual compounding for the same interest rate and time.

What's Next?

Review Tomorrow

In 24 hours, try to write down the difference between the formulas for annual growth and continuous growth from memory.

Practice Activity

Find a local bank's savings interest rate and calculate how long it would take to double your money using the formula $2P = Pe^{rt}$ .

Browse

Browse

Exponential Functions and Growth

Learning Objectives

The Anatomy of Exponential Functions

Modeling Real-World Change

The Natural Base $e$ and Continuous Growth

Key Takeaways

Quick Check

What's Next?

Exponential Functions and Growth

Learning Objectives

The Anatomy of Exponential Functions

Modeling Real-World Change

The Natural Base $e$ and Continuous Growth

Key Takeaways

Quick Check

What's Next?