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Warm-Up

One of these three is not like the others:

f(x) = 8·2ˣ        g(x) = 2^(x+3)        h(x) = 2ˣ + 8

Check: 2^(x+3) = 2³·2ˣ = 8·2ˣ — f and g are the same function: a horizontal shift of an exponential is indistinguishable from a vertical stretch. (h is the odd one out — adding 8 moves the asymptote; multiplying by 8 doesn't.) No polynomial shape-shifts this way. This flexibility is exactly what CED topic 2.4 tests, and it's the algebraic heart of everything logarithmic to come.

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Core Concept

The exponential function

f(x) = a·bˣ        a ≠ 0,  b > 0,  b ≠ 1

• a = f(0): the initial value (y-intercept)
• b: the base — the constant factor per unit of input
• b > 1: growth; 0 < b < 1: decay

For a > 0, the graph:

• is always increasing (b > 1) or always decreasing (0 < b < 1) — no turning points, ever
• is always concave up — the rate of change itself grows (or shrinks toward 0) in proportion to the value
• has no extrema and no points of inflection (monotonic + one concavity = nothing to turn, nothing to switch)
• has horizontal asymptote y = 0: lim x→−∞ a·bˣ = 0 for b > 1 (and lim x→+∞ … = 0 for 0 < b < 1), with the other end running to ±∞
• has no zeros: a·bˣ is never 0

If a < 0, the graph flips over the x-axis: still monotonic, concave down, asymptote y = 0. Adding d shifts the asymptote: a·bˣ + d levels off at y = d.

[GRAPH: Three curves. f(x) = 2ˣ rising left-to-right, concave up, hugging y = 0 on the left. g(x) = (1/2)ˣ falling left-to-right, concave up, hugging y = 0 on the right (mirror image). h(x) = 3·2ˣ − 4 rising with dashed horizontal asymptote y = −4, y-intercept (0, −1). All asymptotes dashed and labeled.]

Exponent properties = graph transformations in disguise

The three identities that make exponentials shape-shifters:

b^(x+k) = bᵏ · bˣ        horizontal shift  ≡  vertical dilation
b^(cx)  = (bᶜ)ˣ          horizontal dilation  ≡  change of base
b^(−x)  = (1/b)ˣ         reflection over y-axis  ≡  reciprocal base

Examples of each: - 2^(x+3) = 8·2ˣ — shifting left 3 IS stretching vertically by 8 - 9^(x/2) = (9^(1/2))ˣ = 3ˣ — halving the input speed IS switching base to √9 - 2^(−x) = (1/2)ˣ — growth reflected is decay

Rational exponents evaluate exactly: 16^(3/4) = (16^(1/4))³ = 2³ = 8. Root first (keep numbers small), then power.

The natural base e

e ≈ 2.718 (between 2 and 3) is a specific irrational constant that arises from continuous growth processes; eˣ is just an exponential function with base e — growth (since e > 1), all the properties above apply. Models with e appear from Lesson 12 on; for now, treat e as a number you know the size of.

Why "same factor per unit input" is the definition

Over ANY interval of length 1, f(x+1)/f(x) = b. Over length Δx, the factor is b^(Δx). Exponentials are the functions whose output ratios depend only on input differences — the multiplicative twin of linearity. This proportional-growth property is the CED's preferred characterization and the phrase to use in FRQ justifications: "over equal-length input intervals, the output is multiplied by equal factors."

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Worked Examples

Example 1 (easy) — Read the anatomy 🚫 No-Calc

Problem: For f(x) = 5·(0.8)ˣ + 3: initial value, growth or decay, asymptote, end behavior, concavity.

Solution: f(0) = 5 + 3 = 8. Base 0.8 < 1 → decay (decreasing). Asymptote y = 3 (the +3 lifts it). End behavior: lim x→∞ f(x) = 3, lim x→−∞ f(x) = ∞. Concave up (a = 5 > 0). Range: (3, ∞).

Interpretation: Five features, zero computation — the formula is the graph.

Example 2 (medium) — Same function, two costumes 🚫 No-Calc

Problem: Show that g(x) = 4^(x/2 + 1) is an exponential function of the form a·bˣ, identifying a and b.

Solution:

4^(x/2 + 1) = 4¹ · 4^(x/2) = 4 · (4^(1/2))ˣ = 4·2ˣ

So a = 4, b = 2. Check x = 2: original 4^(2) = 16; rewritten 4·4 = 16 ✓

Interpretation: Peel constants out of the exponent (product property), then absorb coefficients of x into the base (power property). Every exponential expression reduces to a·bˣ.

Example 3 (medium) — Find a and b from two points 🚫 No-Calc

Problem: f(x) = a·bˣ passes through (0, 6) and (2, 54), with b > 0. Find the formula.

Solution: a = f(0) = 6. Then 6b² = 54 → b² = 9 → b = 3 (base must be positive).

f(x) = 6·3ˣ

Interpretation: The y-intercept hands you a; any second point pins b via a root. With non-zero-based points, divide instead: f(5)/f(3) = b² — ratios cancel a.

Example 4 (AP-style) — Proportional growth reasoning 🚫 No-Calc

Problem: A quantity Q(t) = a·bᵗ satisfies Q(3) = 40 and Q(7) = 640. Find Q(9), without finding a.

Solution: From t = 3 to 7 (4 steps): b⁴ = 640/40 = 16 → b = 2 (b > 0). From t = 7 to 9 (2 steps): Q(9) = Q(7)·b² = 640·4 = 2560.

Interpretation: Exponentials are ratio machines — travel between known points by multiplying. The initial value is often unnecessary (and the exam writes questions where solving for it wastes your time).

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Common Mistakes

1. Calling (1/2)ˣ "concave down" because it decreases. Decay curves fall but bend upward (they flatten toward the asymptote). For a > 0, exponentials are always concave up. Fix: sketch the hug toward y = 0.
2. Claiming an exponential has a zero. a·bˣ never equals 0 — the graph approaches the asymptote, never arrives. Fix: "no zeros; horizontal asymptote y = 0 (or y = d)."
3. Adding exponents across different bases. 2ˣ·3ˣ is 6ˣ (bases multiply when exponents match), but 2ˣ·2ʸ = 2^(x+y) (exponents add when bases match). Mixing the two rules creates 5ˣ-style nonsense. Fix: check what matches before applying any property.
4. Rational exponents in the wrong order. 16^(3/4) as (16³)^(1/4) is technically equal but forces you through 4096. Fix: root first, power second.
5. b = ±3 from b² = 9. Bases are positive by definition (b > 0, b ≠ 1). Fix: discard negative roots automatically.

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Practice Problems

12. (FRQ-style) 🚫 Q(t) = a·bᵗ models a bacteria count, with Q(2) = 200 and Q(5) = 1600. (i) Find b, then find a. (ii) Rewrite Q(t) in an equivalent form whose base is 8, and interpret what that base tells you about the count over every 3-hour interval. (iii) Find Q(8) without using a.

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FRQ Practice — Task Model: Symbolic Manipulations (FRQ 3 style) 🚫 No-Calc

(a) The functions f(x) = 8·4^(x/2) and g(x) = 2^(x+3) are claimed to be identical. (i) Rewrite f(x) in the form a·bˣ with the smallest possible integer base. (ii) Rewrite g(x) in the same form, and state whether the claim is true.

(b) Solve for x exactly: 27^(x−1) = 9^(2x) (express both sides with a common base first).

(c) The function h(x) = 5·3ˣ satisfies h(x + 2) = k·h(x) for all x. (i) Find k. (ii) Explain what this equation says about outputs of h over input intervals of length 2.

Model Response & Rubric (6 points)

(a) [2 pts] (i) [1 pt] f(x) = 8·(4^(1/2))ˣ = 8·2ˣ. (ii) [1 pt] g(x) = 2³·2ˣ = 8·2ˣ. The claim is true — both equal 8·2ˣ.

(b) [2 pts] Common base 3: 3^(3(x−1)) = 3^(4x) [1 pt] → 3x − 3 = 4x → x = −3 [1 pt]. Check: 27⁻⁴ = 3⁻¹² and 9⁻⁶ = 3⁻¹² ✓

(c) [2 pts] (i) [1 pt] h(x + 2) = 5·3^(x+2) = 9·5·3ˣ = 9·h(x) → k = 9. (ii) [1 pt] Whenever the input increases by 2, the output is multiplied by 9 (= 3²): exponential functions multiply outputs by equal factors over equal-length input intervals.

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Show answer key & explanations

(g) Answer Key

1. (A). 3·2⁴ = 3·16 = 48. (B) uses 2³; (C) computes 3⁴; (D) computes (3·2)·... base confusion.

2. (C). a > 0 with 0 < b < 1: decay → decreasing. (B) is increasing (negative a flips the decay into a rising curve toward 0); (A)/(D) grow.

3. (B). The −2 shifts the asymptote from y = 0 to y = −2. (C) forgets the shift; (A)/(D) grab wrong constants.

4. (D). 2^(x+3) = 2³·2ˣ = 8·2ˣ. (B) turns multiplication into addition; (C) multiplies bases.

5. (A). (9^(1/2))ˣ = 3ˣ. (B) computes 9² instead of 9^(1/2); (D) divides the output, not the exponent.

6. (C). Base 1/2 → decreasing; a = 5 > 0 → concave up (falls while flattening). (A) is the classic concavity trap.

7. (B). 16^(1/4) = 2, then 2³ = 8. (A) computes 16·(3/4); (C) computes 16 ÷ ... wrong operations; (D) is 16^(3/2).

8. (D). a = 6; 6b² = 54 → b² = 9 → b = 3 (bases are positive). (C) forgets the base restriction; (A) reports b².

9. (A). Monotonic (no extrema) + always concave up (no inflection). (C) fails — the infimum 0 is never attained; (D) fails — no zeros.

10. (C). e ≈ 2.718: between 2 and 3. (A) is an approximation of π.

11. (B). 5^(x+2x) = 5^(3x) = (5³)ˣ = 125ˣ. (A) is 5^(2x); (C) multiplies exponents; (D) adds bases.

12. (FRQ-style, 6 points) (i) [2 pts] b³ = Q(5)/Q(2) = 1600/200 = 8 → b = 2. Then a·2² = 200 → a = 50. (ii) [2 pts] Q(t) = 50·2ᵗ = 50·(2³)^(t/3) = 50·8^(t/3). The base 8 shows that the count is multiplied by 8 over every 3-hour interval (consistent with doubling each hour: 2³ = 8). (iii) [2 pts] From t = 5, three steps of doubling: Q(8) = Q(5)·2³ = 1600·8 = 12,800.

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🎯 Exam tip: When two exponential expressions are compared or equated, the entire game is "make the bases match." Powers of 2 (2, 4, 8, 16, 32, 64), powers of 3 (3, 9, 27, 81, 243), and powers of 5 cover nearly every no-calc item — recognize them on sight.