Unit 7: Exponential and General Functions

Topic 1

Exponential Growth and Decay

Exponential functions model situations where a quantity changes by a constant percentage rate over equal time intervals. They have the general form:

y = a \, b^x,

where:

$a$ is the initial value (when $x = 0$ ),
$b$ is the growth or decay factor,
$x$ is the time or independent variable.

\begin{cases} b > 1 & \text{Exponential Growth} \\ 0 < b < 1 & \text{Exponential Decay} \end{cases}

Each increase of 1 unit in

x

multiplies the value of

y

b

, not adds to it. This is the key difference from linear functions, which change by a constant amount.

The percentage rate of change can be written as:

b = 1 + r \quad \text{for growth}, \qquad b = 1 - r \quad \text{for decay},

where

r

is the rate expressed as a decimal.

Exponential Growth Example Population, compound interest, or bacteria doubling over time follow growth patterns.

P(t) = P_0 (1 + r)^t

Exponential Decay Example Radioactive decay or depreciation of value follow decay patterns.

A(t) = A_0 (1 - r)^t

Core Skills

Identify $a$ , $b$ , and the growth/decay rate $r$ .
Distinguish between linear (additive) and exponential (multiplicative) change.
Write exponential models from verbal descriptions or data.
Evaluate exponential expressions for given time intervals.
Interpret real-world meanings of constants $a$ , $b$ , and $r$ .

Example 1: Identifying Growth or Decay

Determine whether each function represents growth or decay:

\text{(a) } y = 200(1.05)^x, \quad \text{(b) } y = 500(0.92)^x.

(a)

b = 1.05 > 1

→ Growth (5% increase per unit). (b)

b = 0.92 < 1

→ Decay (8% decrease per unit). Answer: (a) Growth; (b) Decay.

Example 2: Writing an Exponential Model

A population of 1000 bacteria doubles every 4 hours.

P(t) = 1000(2)^{t/4}.

Each 4-hour period multiplies the amount by 2. Interpretation:

a = 1000

, doubling time = 4 hours.

Example 3: Finding the Growth Rate

An investment grows according to

A = 500(1.06)^t.

Here

r = 0.06,

so growth rate = 6% per time unit. After 5 years:

A = 500(1.06)^5 = 500(1.3382) = 669.1.

\boxed{A = \$669.10}

Example 4: Exponential Decay Model

A car’s value is modeled by

V = 30{,}000(0.85)^t,

where

t

is years after purchase. Each year, it retains 85% of its previous value → 15% loss annually. After 3 years:

V = 30{,}000(0.85)^3 = 30{,}000(0.614125) = 18{,}423.75.

\boxed{V = \$18{,}423.75}

Example 5: Comparing Two Models

Compare

y = 100(1.03)^x

and

y = 100(0.97)^x.

The first model grows 3% each period; the second decays 3%. If

x = 10

y_1 = 100(1.03)^{10} = 134.4, \quad y_2 = 100(0.97)^{10} = 73.7.

Interpretation: Growth doubles faster than decay reduces.

Example 6: Graph Behavior

A growth curve (

b > 1

) rises rapidly to the right, while a decay curve (

0 < b < 1

) decreases toward 0. Key features:

Both pass through $(0, a)$ .
Growth curves rise to the right.
Decay curves fall to the right.
The x-axis ( $y = 0$ ) is a horizontal asymptote.

Key Takeaways

Exponential growth multiplies by a constant factor $b > 1$ .
Exponential decay multiplies by $0 < b < 1$ .
Rate $r$ links to $b$ through $b = 1 \pm r.$
Graphs approach but never reach 0 (asymptote at $y = 0$ ).
Recognizing $b$ and interpreting context are key SAT skills.