Exponential Function Manipulation

Part 1 · Hook

Why this matters

A streaming service reports that subscribers double every month. A news site rewrites the same model as "multiplied by 8 every quarter" and another as "up about 19% a week" (taking a month as 4 weeks). All three are the same function, written in different equivalent exponential forms.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. Simplify $2^3 \cdot 2^x$ .

$2^{x + 3}$
$4^{x + 3}$
$2^{3x}$
$6^x$

Show the answer

Same base: add the exponents.

Correct: $2^{x + 3}$ :
$4^{x + 3}$ :
$2^{3x}$ :
$6^x$ :

2. The graph of $g(x) = f(x + 2)$ is the graph of $f$ moved how?

$2$ units left
$2$ units right
$2$ units up
$2$ units down

Show the answer

$g(-2) = f(0)$ : the graph moves left.

Correct: $2$ units left:
$2$ units right:
$2$ units up:
$2$ units down:

Part 4 · See it

See it first

The graph of y = 2^x through (0, 1) and the graph of h(x) = 2^(x + 3) through (-3, 1) and (0, 8): the second is the first moved 3 units left, and also the first with every output multiplied by 8. — $h(x) = 2^{x + 3}$ is $2^x$ shifted $3$ left, and it is also $2^x$ stretched vertically by $2^3 = 8$ : one graph, two descriptions. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

Add to the input: $b^{x + k}$ by the product rule, $b^k \cdot b^x$ : a horizontal translation is a vertical dilation by $b^k$
Multiply the input: $b^{cx}$ by the power rule, $(b^c)^x$ : a horizontal dilation is a change of base
Negate the input: $b^{-x}$ $\left(\frac{1}{b}\right)^x$ : growth reflected across the $y$ -axis is decay

Part 6 · Key ideas

Key ideas

$b^{x + k} = b^k b^x$ : for an exponential function, shifting the graph horizontally is the same as stretching it vertically.
$b^{cx} = (b^c)^x$ and $b^{x/n} = (b^{1/n})^x$ : a horizontal dilation is the same as changing the base.
Use the rules on whole exponents: $4^{x + 1}$ $= 2^{2(x + 1)}$ $= 2^{2x + 2}$ , never $2^{2x + 1}$ .

Part 7 · Misconception

A common mistake

The wrong idea: $2^{x + 3} = 2^x + 2^3$ .

What actually happens: Adding exponents goes with multiplying powers: $2^{x + 3}$ $= 2^x \cdot 2^3$ $= 8(2)^x$ . Check with $x = 1$ : $16$ , not $10$ .

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Graph

The graphs of $y = 2^x$ and $h(x) = 2^{x + 3}$

1. Rewrite $h(x) = 2^{x + 3}$ in the form $a(2)^x$ .

Type an expression for $h(x) =$ .

Show the answer

Product rule backward: $2^{x + 3}$ $= 2^x \cdot 2^3$ $= 8(2)^x$ . The graph shows it: $h(0) = 8$ .

Answer: $h(x) =$ $8(2)^{x}$

2. The graph of $h(x) = 2^{x + 3}$ is the graph of $y = 2^x$ shifted $3$ units left. Which other transformation of $y = 2^x$ gives the same graph?

A vertical shift up $3$
A vertical dilation by a factor of $3$
A vertical dilation by a factor of $8$
A horizontal dilation by a factor of $8$

Show the answer

$2^{x + 3} = 8 \cdot 2^x$ , so every output of $2^x$ is multiplied by $8$ . For an exponential function, a horizontal translation is the same as a vertical dilation.

A vertical shift up $3$ : $2^x + 3$ has asymptote $y = 3$ ; $h$ keeps $y = 0$ .
A vertical dilation by a factor of $3$ : The factor is $2^3$ , not $3$ .
Correct: A vertical dilation by a factor of $8$ : Shift left $3$ = multiply outputs by $2^3$ .
A horizontal dilation by a factor of $8$ : A horizontal dilation would change the base, not multiply the outputs.

Table

Values of an exponential function $g$

Selected values of the exponential function g
$x$	0	1	2	3
$g(x)$	5	20	80	320

3. The table fits $g(x) = 5(4)^x$ . Rewrite $g(x)$ with base $2$ , in the form $a(2)^{kx}$ .

Type an expression for $g(x) =$ .

Show the answer

$4 = 2^2$ , so $4^x = (2^2)^x = 2^{2x}$ and $g(x) = 5(2)^{2x}$ . A horizontal dilation of $5(2)^x$ by a factor of $\frac{1}{2}$ gives the same graph as changing the base to $4$ .

Answer: $g(x) =$ $5(2)^{2x}$

4. Rewrite $\left(\frac{1}{3}\right)^x$ as a power of $3$ .

Type an expression.

Show the answer

$\frac{1}{3} = 3^{-1}$ , so $\left(\frac{1}{3}\right)^x$ $= (3^{-1})^x$ $= 3^{-x}$ . Decay with base $\frac{1}{3}$ is growth with base $3$ reflected across the $y$ -axis.

Answer: $3^{-x}$

5. Write $\frac{4^{x + 1}}{2^x}$ as a single power of $2$ .

Type an expression.

Show the answer

$4^{x + 1}$ $= (2^2)^{x + 1}$ $= 2^{2x + 2}$ . Dividing powers of $2$ subtracts exponents: $\frac{2^{2x + 2}}{2^x}$ $= 2^{2x + 2 - x}$ $= 2^{x + 2}$ .

Answer: $2^{x + 2}$

6. A student writes $2^{x + 3} = 2^x + 8$ . Which check shows the error?

At $x = 0$ : both sides equal $9$ , so the forms agree
At $x = 3$ : $2^6 = 64$ and $2^3 + 8 = 64$ , so the forms agree
At $x = 1$ : $2^4 = 8$ and $2^1 + 8 = 10$
At $x = 1$ : $2^4 = 16$ , but $2^1 + 8 = 10$

Show the answer

One input with different values is enough to show two forms are not equivalent. The correct rule is $2^{x + 3}$ $= 2^x \cdot 2^3$ $= 8 \cdot 2^x$ : adding to the exponent multiplies.

At $x = 0$ : both sides equal $9$ , so the forms agree: $2^{0 + 3} = 8$ , not $9$ ; the forms differ at $x = 0$ too.
At $x = 3$ : $2^6 = 64$ and $2^3 + 8 = 64$ , so the forms agree: $2^3 + 8 = 16$ , not $64$ .
At $x = 1$ : $2^4 = 8$ and $2^1 + 8 = 10$ : $2^4 = 16$ , not $8$ ; a check needs correct values.
Correct: At $x = 1$ : $2^4 = 16$ , but $2^1 + 8 = 10$ : $16 \ne 10$ , so the forms are not equivalent.

Part 9 · Summary

Summary

Exponent rules rewrite exponential functions:

b^{x + k} = b^k b^x

(a horizontal shift is a vertical dilation),

b^{cx} = (b^c)^x

(a horizontal dilation is a new base) and

b^{-x} = \left(\frac{1}{b}\right)^x

. Equivalent forms make different features easy to read.

Part 10 · Up next

What comes next

Part 11 · Connections

Why this matters

What this builds on

Quick check before you start

See it first

How it works, step by step

Key ideas

A common mistake

Check yourself

The graphs of y=2xy = 2^x and h(x)=2x+3h(x) = 2^{x + 3}

Values of an exponential function gg

Summary

What comes next

Connections

The graphs of $y = 2^x$ and $h(x) = 2^{x + 3}$

Values of an exponential function $g$