Unit 2 · Topic 2.4 · about 30 minutes · 7 parts · 13 scored checks, 1 of them multiple choice

Exponential Function Manipulation

Built from College Board's course framework · Every answer key independently checked

The graph of y = 2 to the x, dashed, and the graph of g(x) = 2 to the (x + 2), solid, which is the same curve as 4 times 2 to the x. The point (0, 1) on the dashed graph lands on the solid graph two ways: moved 2 units left to (-2, 1), or stretched up by a factor of 4 to (0, 4).

Part 1 Think first

Two models for one colony

Guess before you learn. A wrong guess still makes the answer stick.

Predict first

A yeast colony doubles every hour. Two students model it, with $t$ in hours and the count in thousands of cells. Ana writes $A (t) = 2^{t + 3}$ . Ben writes $B (t) = 8 \cdot 2^{t}$ . Without a calculator, which is true?

In short. Manipulating an exponential function means using the properties of exponents to rewrite it in an equivalent form that shows something the first form hid.

The product property, $b^{m} b^{n} = b^{m + n}$ , makes every horizontal translation $b^{x + k}$ the same as the vertical dilation $b^{k} \cdot b^{x}$ .
The power property, $(b^{m})^{n} = b^{m n}$ , makes every horizontal dilation $b^{c x}$ the same as a change of base, $(b^{c})^{x}$ .
A negative exponent gives a reciprocal, $b^{- n} = \frac{1}{b^{n}}$ , and an exponent of $\frac{1}{k}$ gives the $k$ th root of $b$ , when it exists.

Part 2 Learn it

Three short ideas, each checked as you go

2a. Product property: a translation is a dilation

When you multiply two powers of the same base, the exponents add: $b^{m} b^{n} = b^{m + n}$ . That is the product property for exponents. Read it from right to left and it splits an exponent apart.

Take $f (x) = 3^{x + 2}$ . Adding 2 to the input is a horizontal translation, 2 units left. Now split the exponent:

3^{x + 2} = 3^{2} \cdot 3^{x} = 9 \cdot 3^{x}

Multiplying every output by 9 is a vertical dilation. So one graph is both: the graph of $3^{x}$ moved 2 units left, or made 9 times as tall. In general, $b^{x + k} = b^{k} \cdot b^{x} = a b^{x}$ with $a = b^{k}$ . Every horizontal translation of an exponential function is a vertical dilation, and the diagram shows it for $2^{x + 2} = 4 \cdot 2^{x}$ .

Which form you want depends on the question. $9 \cdot 3^{x}$ shows the $y$ -intercept, 9, at a glance. $3^{x + 2}$ shows where the graph of $3^{x}$ went.

Quick check · Calculate

No calculator. The function $f (x) = 2^{x + 5}$ can be written as $f (x) = a \cdot 2^{x}$ . What is $a$ ?

Your answer

2b. Power property: a horizontal dilation is a new base

Raise a power to a power and the exponents multiply: $(b^{m})^{n} = b^{m n}$ . That is the power property for exponents.

Read it backward on $g (x) = 2^{3 x}$ . Multiplying the input by 3 is a horizontal dilation by a factor of $\frac{1}{3}$ . But

2^{3 x} = (2^{3})^{x} = 8^{x}

so the same graph is an exponential function with a new base, 8. Each step of 1 in $x$ now multiplies the output by 8, not by 2. In general, $b^{c x} = (b^{c})^{x}$ for any constant $c \neq 0$ : every horizontal dilation of an exponential function is a change of base.

Keep the two properties apart, because they look alike. A number added in the exponent splits off as a factor in front: $2^{x + 3} = 8 \cdot 2^{x}$ . A number multiplying the exponent goes onto the base: $2^{3 x} = 8^{x}$ . Same 8, different place, and the two functions are not equal. At $x = 2$ the first is 32 and the second is 64.

Quick check · Match

No calculator. Tap a function, then the equivalent form.

2c. Negative exponents and unit fraction exponents

Two more properties let an exponent go below zero or sit between whole numbers.

The negative exponent property: $b^{- n} = \frac{1}{b^{n}}$ . A negative exponent means a reciprocal, never a negative number. $2^{- 3} = \frac{1}{8}$ , which is positive, like every output of $b^{x}$ when $b > 0$ .

An exponent of $\frac{1}{k}$ , where $k$ is a natural number, is an exponential unit fraction, and it takes a root: $b^{\frac{1}{k}}$ is the $k$ th root of $b$ , when that root exists. So $25^{\frac{1}{2}} = \sqrt{25} = 5$ and $8^{\frac{1}{3}} = \sqrt[3]{8} = 2$ . The words "when it exists" matter: $(- 4)^{\frac{1}{2}}$ has no real value. The base of an exponential function is positive, so you will not meet that problem there.

These combine with the power property. $27^{- \frac{x}{3}} = {(27^{- \frac{1}{3}})}^{x} = {(\frac{1}{3})}^{x}$ : a cube root, then a reciprocal. A function that looked like growth, base 27, turns out to be decay with base $\frac{1}{3}$ .

Quick check · Sort it

No calculator. Tap an expression, then its value.

1/4

1/2

Part 3 Watch it done

Rewriting into a form you can read

Watch each step, then do the last one yourself.

Rewrite $f (x) = 4^{\frac{x - 1}{2}}$ in the form $f (x) = a b^{x}$ , where $a$ and $b$ are constants.

Step 1 · Split the exponent
$\frac{x - 1}{2} = \frac{x}{2} - \frac{1}{2}$ , so the product property gives $f (x) = 4^{\frac{x}{2}} \cdot 4^{- \frac{1}{2}}$ . The second factor has no $x$ in it. It is a constant, and it will become $a$ .
Step 2 · Change the base
$4^{\frac{x}{2}} = {(4^{\frac{1}{2}})}^{x}$ by the power property, and $4^{\frac{1}{2}} = \sqrt{4} = 2$ . So $4^{\frac{x}{2}} = 2^{x}$ , and $f (x) = 4^{- \frac{1}{2}} \cdot 2^{x}$ . The base is $b = 2$ .
Why this step?
Why is $4^{\frac{x}{2}}$ equal to $2^{x}$ , and not $2^{\frac{x}{2}}$ ?
Dividing the exponent by 2 is multiplying it by $\frac{1}{2}$ , and the power property moves that $\frac{1}{2}$ onto the base as a square root. The root is taken once, of the base, and the $x$ stays whole. Check at $x = 2$ : $4^{\frac{2}{2}} = 4$ and $2^{2} = 4$ , while $2^{\frac{2}{2}} = 2$ .
Your step · Calculate

No calculator. Finish it. In $f (x) = a b^{x}$ , what is $a$ ? Give a decimal.

Your answer

Part 4 Try it

Make a translation into a dilation

Lab

Function Grapher

Use the exponential family, $f (x) = 2^{x}$ . The lab writes $g (x) = a f (b (x + c)) + d$ , so $c$ translates the graph sideways and $a$ dilates it vertically. Work out your answer with the product property first, then use the sliders to check that the two graphs land on each other.

Open the full Function Grapher lab

Try it · Calculate

No calculator. The lab graphs $g (x) = a \cdot 2^{b (x + c)} + d$ for the exponential family. Set $a = 1$ , $b = 1$ , $d = 0$ and $c = - 2$ , so the colored graph is $g (x) = 2^{x - 2}$ . Now set $c$ back to 0. What value of $a$ gives the same graph? Give a decimal, then check with the slider.

Your answer

Part 5 Explain it

Write it the way the exam asks

Nothing you type leaves this page. Check your answer against the model and each scoring point.

Free response is 37.5% of the exam score. This one is shaped like Question 4, Symbolic Manipulations.

Free-response practice · Question 4 style

No calculator. Let $g (x) = 9^{\frac{x}{2} + 1}$ . (A) Rewrite $g (x)$ in the form $a b^{x}$ , where $a$ and $b$ are constants, and name the property of exponents you use at each step. (B) The graph of $g$ is a horizontal translation of the graph of $y = 3^{x}$ . Find the translation, and explain why it is also a vertical dilation.

Your answer stays on this page. It is never sent or saved.

Model answer

(A) By the product property, $9^{\frac{x}{2} + 1} = 9^{\frac{x}{2}} \cdot 9^{1}$ . By the power property, $9^{\frac{x}{2}} = {(9^{\frac{1}{2}})}^{x}$ , and $9^{\frac{1}{2}} = \sqrt{9} = 3$ . So $g (x) = 9 \cdot 3^{x}$ . (B) Since $9 = 3^{2}$ , $g (x) = 3^{2} \cdot 3^{x} = 3^{x + 2}$ , which is the graph of $y = 3^{x}$ translated 2 units left. By the product property, $3^{x + 2} = 3^{2} \cdot 3^{x} = 9 \cdot 3^{x}$ , so the same graph is the graph of $y = 3^{x}$ dilated vertically by a factor of 9.

Score your own answer. This is a self-check, not a grade. Tick each point your answer makes.

Splits the exponent into $9^{\frac{x}{2}} \cdot 9$ and names the product property.
Rewrites $9^{\frac{x}{2}}$ as ${(9^{\frac{1}{2}})}^{x} = 3^{x}$ , naming the power property and the square root.
States $g (x) = 9 \cdot 3^{x}$ , so $a = 9$ and $b = 3$ .
Finds the translation, 2 units left, and uses $3^{x + 2} = 3^{2} \cdot 3^{x}$ to explain why it is a vertical dilation by 9.

Part 6 Check yourself

Three forms of one function

The three case questions use the same three forms.

Three students each wrote a form of the same function $f$ . All three forms are equivalent.

Three equivalent forms of $f (x)$
Form	$f (x)$
P	$2 \cdot 2^{x}$
Q	$\frac{1}{2} \cdot 4^{\frac{x}{2} + 1}$
R	${(\frac{1}{2})}^{- x - 1}$

Case question 1 · Spot the error

No calculator. A fourth student rewrites Form R with base 2. Tap the first line with a mistake.

Case question 2 · Calculate

No calculator. What is $f (- 4)$ ? Give a decimal.

Your answer

Case question 3 · Multiple choice

No calculator. The graph of $f$ is a horizontal translation of the graph of $y = 2^{x}$ . Rewrite one of the forms to find which translation.

The rest stand on their own and do not use the case.

Check yourself · Put it in order

No calculator. Start from $27^{\frac{2 x}{3}}$ . Put the expressions in order so that each one rewrites the one before it, ending in the form $b^{x}$ .

Check yourself · Decide and justify

No calculator. Let $g (x) = 5^{x - 2}$ . A classmate says the graph of $g$ is the graph of $y = 5^{x}$ dilated vertically. Is she right?

Which reason supports your answer?

Check yourself · Fill the statement

No calculator. Find $32^{- \frac{1}{5}}$ . Tap a blank, then tap a word or a value.

The $\frac{1}{5}$ takes the of 32. The negative sign takes the of that, so $32^{- \frac{1}{5}}$ is .

Part 7 Keep it

Bring back earlier ideas

Pulling an older idea back up is what makes it last.

From 1.12 Transformations of Functions · Calculate

No calculator. The point $(6, 2)$ is on the graph of $f$ . The corresponding point on the graph of $g (x) = 3 f (2 x)$ is $(p, q)$ . What is $p$ ?

Your answer

From 1.12 Transformations of Functions · Sort it

No calculator. Tap a transformation of $f$ , then whether it changes the graph vertically or horizontally.

Vertically

Horizontally

In three lines

Product property: $b^{x + k} = b^{k} \cdot b^{x}$ , so a horizontal translation of an exponential function is a vertical dilation.
Power property: $b^{c x} = (b^{c})^{x}$ , so a horizontal dilation of an exponential function is a change of base.
$b^{- n} = \frac{1}{b^{n}}$ , and $b^{\frac{1}{k}}$ is the $k$ th root of $b$ when it exists.

Vocabulary

Product property for exponents: $b^{m} b^{n} = b^{m + n}$ : multiplying powers of the same base adds the exponents.
Power property for exponents: $(b^{m})^{n} = b^{m n}$ : raising a power to a power multiplies the exponents.
Negative exponent property: $b^{- n} = \frac{1}{b^{n}}$ : a negative exponent gives the reciprocal of the positive power.
Exponential unit fraction: An exponent of $\frac{1}{k}$ , where $k$ is a natural number; $b^{\frac{1}{k}}$ is the $k$ th root of $b$ , when it exists.
Horizontal translation and vertical dilation: For an exponential function these can be the same graph: $b^{x + k}$ moves the graph of $b^{x}$ sideways and equals $b^{k} \cdot b^{x}$ , which stretches it vertically.

Course alignment, for teachers

AP Precalculus topic 2.4, Unit 2: Exponential and Logarithmic Functions.

Skill 1.B: Express functions, equations, or expressions in analytically equivalent forms that are useful in a given mathematical or applied context.
Skill 3.A: Describe the characteristics of a function with varying levels of precision, depending on the function representation and available mathematical tools.

This lesson maps all 4 essential knowledge statements of the topic to the part that teaches each one: 4 are checked by a scored question. The map is in the lesson file.

Two models for one colony

Three short ideas, each checked as you go

2a. Product property: a translation is a dilation

2b. Power property: a horizontal dilation is a new base

2c. Negative exponents and unit fraction exponents

Rewriting into a form you can read

Make a translation into a dilation

Function Grapher

Write it the way the exam asks

Three forms of one function

Bring back earlier ideas

Vocabulary

Labs that use this topic