AP® Precalculus review sheet from Aim for Five (aimforfive.com/precalc/units/2/2-4)
Unit 2 · Topic 2.4
2.4 Exponential Function Manipulation
Exponent rules let you rewrite exponential expressions in equivalent forms. Each rule also has a graph meaning: for exponential functions, a horizontal shift is the same as a vertical stretch, and a horizontal stretch is the same as a change of base.
Key terms
- product property
- power property
- negative exponent
- rational exponent
- equivalent forms
The rules
These hold for b > 0 and any real exponents m and n:
| Property | Rule | Example |
|---|---|---|
| Product | bᵐ · bⁿ = bᵐ⁺ⁿ | 2³ · 2⁴ = 2⁷ |
| Power | (bᵐ)ⁿ = bᵐⁿ | (3²)⁵ = 3¹⁰ |
| Negative exponent | b⁻ⁿ = 1/bⁿ | 5⁻² = 1/25 |
| Unit fraction exponent | b^(1/k) = kth root of b | 8^(1/3) = 2 |
Rational exponents
Combine the power rule with roots: b^(m/k) is the kth root of b, raised to the m. For example, 27^(2/3) = (cube root of 27)² = 3² = 9, and 16^(3/4) = 2³ = 8.
A unit fraction exponent like b^(1/2) is the square root of b, so 9^(1/2) = 3. This is defined whenever the root exists; for b > 0 it always does.
Negative and fractional exponents combine: 8^(−2/3) = 1/8^(2/3) = 1/(cube root of 8)² = 1/4. Handle the negative sign with a reciprocal, take the root, then raise to the power. Taking the root first keeps the numbers small.
Horizontal shift = vertical dilation
By the product property, b^(x + k) = bᵏ · bˣ. So shifting the graph of bˣ horizontally by −k units gives exactly the same graph as stretching it vertically by the factor bᵏ.
Example: 2^(x + 3) = 2³ · 2ˣ = 8 · 2ˣ. Shifting 2ˣ left 3 units is the same as multiplying its outputs by 8.
It works in reverse too. 9 · 3ˣ = 3² · 3ˣ = 3^(x + 2), so stretching 3ˣ vertically by 9 is the same as shifting it left 2. Even 5 · 2ˣ is a left shift of 2ˣ, by the power of 2 that equals 5. That power is a little more than 2, and logs (topic 2.9) give it exactly.
Horizontal dilation = change of base
By the power property, b^(cx) = (bᶜ)ˣ. So stretching or squeezing an exponential graph horizontally just changes its base.
Example: 4^(x/2) = (4^(1/2))ˣ = 2ˣ, and 2^(3x) = (2³)ˣ = 8ˣ.
And by the negative exponent property, b^(−x) = (1/b)ˣ. Reflecting 2ˣ over the y-axis gives (1/2)ˣ, turning growth into decay.
Why rewrite at all?
Different forms answer different questions. A form like a · bˣ shows the initial value and growth factor right away. Rewriting 3 · 2^(x − 4) as (3/16) · 2ˣ shows the value at x = 0 is 3/16. Rewriting 9ˣ as 3^(2x) lets you compare it with 3ˣ or solve an equation where both sides have base 3.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Rewriting in a · bˣ form
Rewrite f(x) = 3 · 2^(x − 4) in the form a · bˣ, and describe what this says about the graph.
Show the solutionHide the solution
- Step 1: Use the product property: 2^(x − 4) = 2ˣ · 2⁻⁴ = 2ˣ/16.
- Step 2: So f(x) = 3 · (1/16) · 2ˣ = (3/16) · 2ˣ.
- Step 3: The original form is 3 · 2ˣ shifted right 4. The new form is 3 · 2ˣ shrunk vertically by a factor of 1/16. They are the same graph.
Answer: f(x) = (3/16) · 2ˣ = 0.1875 · 2ˣ.
- Example 2
Changing the base
Rewrite each in the form a · bˣ: (a) g(x) = 5 · 8^(x/3), (b) h(x) = 4^(x/2 + 1), (c) k(x) = (1/9)^(−x/2).
Show the solutionHide the solution
- Step 1: (a) 8^(x/3) = (8^(1/3))ˣ = 2ˣ, so g(x) = 5 · 2ˣ.
- Step 2: (b) 4^(x/2 + 1) = 4^(x/2) · 4¹ = 4 · (4^(1/2))ˣ = 4 · 2ˣ.
- Step 3: (c) (1/9)^(−x/2) = ((1/9)^(−1/2))ˣ. Now (1/9)⁻¹ = 9 and 9^(1/2) = 3, so this is 3ˣ.
Answer: (a) 5 · 2ˣ; (b) 4 · 2ˣ; (c) 3ˣ.
- Example 3
Trap: multiplying the base by the coefficient
A student simplifies 2 · 3ˣ to 6ˣ. Test whether that's correct.
Show the solutionHide the solution
- Step 1: Try x = 2. 2 · 3² = 2 · 9 = 18, while 6² = 36. They aren't equal.
- Step 2: The exponent applies only to the base 3. The coefficient 2 is multiplied once, after the power is computed.
- Step 3: 2 · 3ˣ is already in simplest a · bˣ form.
Answer: Not correct: 2 · 3ˣ ≠ 6ˣ (for example, 18 ≠ 36 at x = 2).
Common mistakes
- Combining a coefficient with the base, as in 2 · 3ˣ = 6ˣ. Exponents act only on what they're attached to.
- Adding exponents when you should multiply them: (2³)⁴ = 2¹², not 2⁷.
- Thinking a negative exponent makes a number negative. 2⁻³ = 1/8, which is positive.
On the exam
- No-calculator questions often ask for an equivalent form, such as rewriting a horizontally shifted exponential as a vertical dilation. Show each property you use.
- Rewriting so both sides share a base is a fast way to solve equations like 4ˣ = 8^(x − 1) (both sides become powers of 2).
Connected topics
Videos
Check yourself
4 questions on 2.4 Exponential Function Manipulation. Pick an answer to see if you got it, and why.
Which of the following is equivalent to 9^(x/2 − 1)?
Which of the following is equivalent to 3^(2x + 1)?
The function g(x) = 5 · 2^(x − 3) can also be written as g(x) = k · 2ˣ. What is the value of k?
Which of the following is equivalent to (√5)^(4x) / 5ˣ?
0 of 4 answered