Exponential Functions

Unit 2 · Exponential and Logarithmic Functions

What AP Precalc asks here

An exponential function has the form $f(x) = a \cdot b^x$ , where $a$ is the initial value (the y-intercept) and $b$ is the growth/decay factor. Growth happens when $b > 1$ ; decay when $0 < b < 1$ . The percent rate $r$ relates to $b$ by $b = 1 + r$ for growth or $b = 1 - r$ for decay — confusing these is the most common error on this topic.

Exponential function form

Standard

f(x) = a \cdot b^x \quad (a \neq 0, \; b > 0, \; b \neq 1)

Y-intercept

(0, a)

Horizontal asymptote

y = 0

Percent rate to factor

Growth rate r

b = 1 + r

Decay rate r

b = 1 - r

Caution: A '5% per year growth rate' means

b = 1.05

, not

0.05

. The growth factor includes the original 100%.

Type 1

Identify initial value and growth factor

Read off a and b directly from $f(x) = a \cdot b^x$ . Decide growth or decay by comparing b to 1.

Example 1

For

g(x) = 50 \cdot 0.8^x

, identify the initial value, the growth factor, and the per-step percent change. (A)

a = 50

b = 0.8

, 20% decay per step (B)

a = 0.8

b = 50

, 20% growth per step (C)

a = 50

b = 0.8

, 80% decay per step (D)

a = 50

b = 0.8

, 8% decay per step

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Type 2

Build f(x) from initial value and percent rate

Set $a$ to the initial value and convert the percent rate to $b = 1 + r$ (growth) or $b = 1 - r$ (decay).

Example 2

A population starts at 5,000 and grows 6% per year. Write

P(t)

. (A)

P(t) = 5{,}000 \cdot 0.06^t

(B)

P(t) = 5{,}000 \cdot 1.06^t

(C)

P(t) = 5{,}000 \cdot 6^t

(D)

P(t) = 5{,}000 \cdot 1.6^t

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Type 3

Fit f(x) through two points

If one point has $x = 0$ , use it to read $a$ . Then divide the other equation by $a$ and take an appropriate root to isolate $b$ .

Example 3

f(x) = a \cdot b^x

passes through

(0, 5)

and

(3, 40)

, find

a

and

b

. (A)

a = 5, b = 2

(B)

a = 5, b = 8

(C)

a = 40, b = 2

(D)

a = 5, b = 4

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