📌 Core Idea
The exponent properties you learned in algebra can also be used to manipulate and rearrange exponential functions into equivalent forms — turning horizontal transformations into vertical ones, and vice versa.
📐 Review of Important Exponent Rules
Product Property
bm · bn = bm+n
53 · 58 = 511
2x · 23 = 2x+3
2x · 23 = 2x+3
Power Property
(bm)n = bmn
(53)x = 53x
(2x)3 = 23x
(2x)3 = 23x
Negative Exponent
b−n = 1 / bn
x−3 = 1/x3
x−½ = 1/√x
x−½ = 1/√x
🔄 Horizontal Translation ↔ Vertical Dilation
Using the Product Property
A horizontal shift of an exponential function can always be rewritten as a vertical dilation using the Product Property in reverse.
f(x) = 4x+2
= 4x · 42
= 16 · 4x
= 4x · 42
= 16 · 4x
Important Idea: Every horizontal translation bx+h is equivalent to a vertical dilation a · bx, where a = bh.
✏️ Example 1 — Identify the Horizontal Transformation
Part a
f(x) = 4x+2
Shifted left 2 units
Part b
g(x) = 2x/3
Horizontal dilation by ⅓
Part c
h(x) = 92x
Horizontal dilation by 2
Part d
k(x) = 5x−1
Shifted right 1 unit
✏️ Example 2 — Rewrite as Vertical Dilation (No Horizontal Shift)
f(x)
2x+3
= 23 · 2x = 8 · 2x
g(x)
3x−2
= 3−2 · 3x = ⅑ · 3x
k(x)
4 · 3x+2
= 4 · 9 · 3x = 36 · 3x
🔄 Horizontal Dilation ↔ Base Change
Using the Power Property
A horizontal dilation of an exponential function bcx is equivalent to changing the base using the Power Property.
f(x) = bcx = (bc)x
For example:
y = 92x = (92)x = 81x
✏️ Examples 3 & 4 — Multiple Choice
Which is an equivalent form of y = 2 · 9x?
(A) f(x) = 3x
(B) f(x) = 3 · 9x
(C) f(x) = 18x
✓ (D) f(x) = 2 · (32)x = 2 · 9x
Which is an equivalent form of y = 9 · 4x?
(A) f(x) = 23 · 16x
(B) f(x) = 23 · 16x
✓ (C) f(x) = 9 · 4x = 9 · (22)x
(D) f(x) = 9 · 16x
Card 1 of 4
Concept
What is the Product Property of Exponents?
Tap to reveal answer ✨
Answer
bm · bn = bm+n
When multiplying powers with the same base, add the exponents.
Example: 53 · 58 = 511
Example: 53 · 58 = 511
Topic 2.4
Exponential Manipulation Quiz
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Question 1 of 4
Which of the following is an equivalent form of f(x) = 4x+2 that has a vertical dilation and no horizontal translation?
Question 2 of 4
Using the Power Property, which of the following is equivalent to y = 92x?
Question 3 of 4
Rewrite g(x) = 3x−2 in the form a · 3x. What is the value of a?
Question 4 of 4
Which of the following is an equivalent form of f(x) = 4 · 3x+2 with no horizontal translation?
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