2.4 - Exponential Function Manipulation

Overview

The same exponential function can be written in many ways:

2^{x+3}

and

8 \cdot 2^x

look different, but they're the exact same function. This topic covers the exponent rules that let you switch between forms, and what each rule means for the graph. AP questions often ask you to rewrite an expression in a specific form, like

ab^x

, and that takes exactly these skills.

The product property

b^m b^n = b^{m+n}

Why it works:

b^m

m

factors of

b

and

b^n

n

more, so together there are

m + n

factors. For example,

2^3 \cdot 2^4 = (2 \cdot 2 \cdot 2)(2 \cdot 2 \cdot 2 \cdot 2) = 2^7

What it means for the graph

Apply the product property to a horizontal translation of

y = b^x

b^{x+k} = b^x \cdot b^k = ab^x, \quad \text{where } a = b^k

So every horizontal translation of an exponential function is also a vertical dilation. For example,

2^{x+3} = 2^3 \cdot 2^x = 8 \cdot 2^x

Shifting

y = 2^x

left 3 units gives exactly the same graph as stretching it vertically by a factor of 8. In the figure, the point

(2, 4)

moves left 3 to

(-1, 4)

, and the point

(-1, 0.5)

stretches up by a factor of 8 to the same point

(-1, 4)

A shift to the right works the same way, but it gives a vertical compression:

2^{x-1} = 2^{-1} \cdot 2^x = \frac{1}{2} \cdot 2^x

Related rule: dividing powers with the same base subtracts exponents,

\frac{b^m}{b^n} = b^{m-n}

. It comes from the product property together with negative exponents (below).

The power property

\left(b^m\right)^n = b^{mn}

Why it works:

\left(b^m\right)^n

n

copies of

b^m

multiplied together, and each copy has

m

factors of

b

, for

mn

factors in all. For example,

\left(2^3\right)^2 = 2^3 \cdot 2^3 = 2^6

What it means for the graph

Apply the power property to a horizontal dilation of

y = b^x

b^{cx} = \left(b^c\right)^x, \quad c \neq 0

Since

b^c

is just a constant, this is an exponential function with a new base. So every horizontal dilation of an exponential function is a change of base. For example,

2^{2x} = \left(2^2\right)^x = 4^x

Compressing

y = 2^x

horizontally by a factor of

\frac{1}{2}

gives

y = 4^x

. Every point's

x

-coordinate is cut in half:

(2, 4)

moves to

(1, 4)

, and

(3, 8)

moves to

(1.5, 8)

It works with fractions too:

9^{x/2} = \left(9^{1/2}\right)^x = 3^x

The negative exponent property

b^{-n} = \frac{1}{b^n}

Why it works: by the product property,

b^n \cdot b^{-n} = b^0 = 1

, so

b^{-n}

is the number you multiply

b^n

by to get 1. That's the reciprocal,

\frac{1}{b^n}

A negative exponent does not make the value negative:

2^{-3} = \frac{1}{8}

, not

-8

For graphs, combine it with the power property (using

c = -1

2^{-x} = \left(2^{-1}\right)^x = \left(\tfrac{1}{2}\right)^x

Reflecting

y = 2^x

over the

y

-axis gives

y = \left(\frac{1}{2}\right)^x

. Any growth function reflected over the

y

-axis becomes a decay function, and the new base is the reciprocal of the old one.

Unit fraction exponents and roots

An exponent of the form

\frac{1}{k}

, where

k

is a natural number, means the

k

th root:

b^{1/k} = \sqrt[k]{b}

Why it works: by the power property,

\left(b^{1/k}\right)^k = b^{(1/k) \cdot k} = b^1 = b

. So

b^{1/k}

is the number whose

k

th power is

b

, which is exactly what the

k

th root means. For example,

8^{1/3} = 2

because

2^3 = 8

, and

81^{1/4} = 3

because

3^4 = 81

The root has to exist.

(-8)^{1/3} = -2

is fine, but

(-16)^{1/4}

is not a real number, since no real number to the 4th power is negative. In exponential functions the base is always positive, so this never causes a problem there.

Other fraction exponents combine a root and a power:

b^{m/k} = \left(b^{1/k}\right)^m

. For example,

27^{2/3} = \left(27^{1/3}\right)^2 = 3^2 = 9

Property	Rule	What it means for $y = b^x$
Product	$b^m b^n = b^{m+n}$	A horizontal translation is a vertical dilation: $b^{x+k} = b^k \cdot b^x$
Power	$\left(b^m\right)^n = b^{mn}$	A horizontal dilation is a change of base: $b^{cx} = \left(b^c\right)^x$
Negative exponent	$b^{-n} = \frac{1}{b^n}$	A reflection over the $y$ -axis gives base $\frac{1}{b}$ : $b^{-x} = \left(\frac{1}{b}\right)^x$
Unit fraction	$b^{1/k} = \sqrt[k]{b}$	Rewrites bases with roots, like $4^{x/2} = 2^x$

Example 1: Translation to dilation

Write

f(x) = 5 \cdot 3^{x-2}

in the form

ab^x

f(x) = 5 \cdot 3^x \cdot 3^{-2} = 5 \cdot \frac{1}{9} \cdot 3^x = \frac{5}{9} \cdot 3^x

Answer:

f(x) = \frac{5}{9}(3)^x

, so

a = \frac{5}{9}

and

b = 3

. Shifting

y = 5 \cdot 3^x

right 2 units is the same as multiplying its outputs by

\frac{1}{9}

Example 2: Changing the base

Write each function in the form

ab^x

: (a)

g(x) = 7 \cdot 4^{x/2}

and (b)

h(x) = 10\left(\frac{1}{2}\right)^{-2x}

(a) Use the power property with

c = \frac{1}{2}

g(x) = 7\left(4^{1/2}\right)^x = 7 \cdot 2^x

(b) Use the power property with

c = -2

, then the negative exponent property:

h(x) = 10\left(\left(\tfrac{1}{2}\right)^{-2}\right)^x = 10\left(2^2\right)^x = 10 \cdot 4^x

Answer:

g(x) = 7 \cdot 2^x

and

h(x) = 10 \cdot 4^x

. Both show exponential growth, even though

h

was written with a base less than 1.

Example 3: Evaluating without a calculator

$32^{3/5} = \left(32^{1/5}\right)^3 = 2^3 = 8$

$9^{-1/2} = \frac{1}{9^{1/2}} = \frac{1}{3}$

$16^{-3/4} = \frac{1}{\left(16^{1/4}\right)^3} = \frac{1}{2^3} = \frac{1}{8}$

$\left(\frac{1}{8}\right)^{-2/3} = 8^{2/3} = \left(8^{1/3}\right)^2 = 2^2 = 4$

Take the root first. It keeps the numbers small.

Example 4: Rewriting a combined expression (AP-style)

Rewrite

k(x) = \dfrac{2^{x+3} \cdot 4^{x-1}}{8^{x/3}}

in the form

ab^x

. Does

k

show exponential growth or decay?

Step 1: write every factor with base 2, since

4 = 2^2

and

8 = 2^3

4^{x-1} = \left(2^2\right)^{x-1} = 2^{2x-2} \qquad 8^{x/3} = \left(2^3\right)^{x/3} = 2^x

Step 2: combine with the product and quotient rules.

k(x) = \frac{2^{x+3} \cdot 2^{2x-2}}{2^x} = 2^{(x+3) + (2x-2) - x} = 2^{2x+1}

Step 3: split off the constant and change the base.

2^{2x+1} = 2^1 \cdot 2^{2x} = 2\left(2^2\right)^x = 2 \cdot 4^x

Answer:

k(x) = 2 \cdot 4^x

. Since

a = 2 > 0

and

b = 4 > 1

k

shows exponential growth: its output is multiplied by 4 each time

x

increases by 1. Check with

x = 1

: the original gives

\frac{2^4 \cdot 4^0}{8^{1/3}} = \frac{16}{2} = 8

, and

2 \cdot 4^1 = 8

Common mistakes

Multiplying exponents when you should add. $b^m b^n = b^{m+n}$ , not $b^{mn}$ . Multiplying exponents is for a power of a power.

Combining the coefficient with the base. $3 \cdot 2^x$ is not $6^x$ . The exponent only applies to the 2.

Treating a negative exponent as a negative number. $2^{-3} = \frac{1}{8}$ .

Not distributing the exponent. $4^{x-1} = 2^{2(x-1)} = 2^{2x-2}$ , not $2^{2x-1}$ .

Turning a sum in the exponent into a sum of powers. $2^{x+3} = 2^x \cdot 2^3$ , not $2^x + 2^3$ .

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