Mathematics Advanced • Year 11 • Module 4 • Lesson 1

Introduction to Exponential Functions

Build procedural fluency in identifying, classifying, and evaluating exponential functions of the form y = a^x.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 Complete the definition:

A function of the form y = a^x is called an exponential function provided a > ____ and a ≠ ____.

Q1.2 Fill in the table:

If a > 1, the function shows ____________ ; if 0 < a < 1, the function shows ____________.

Q1.3 State the y-intercept of every basic exponential y = a^x: (____, ____).

Stuck? Revisit lesson § Concept and § Key Terms.

2. Worked example — evaluate f(x) = 2^x at x = −2, 0, 3

Follow each line. Every step has a reason on the right.

Problem. Evaluate f(x) = 2^x at x = −2, x = 0, and x = 3.

Step 1 — Evaluate at x = −2 using the negative-exponent rule.

f(−2) = 2⁻² = 1 / 2² = 1/4

Reason: a⁻ⁿ = 1/aⁿ; the negative exponent gives the reciprocal, not a negative number.

Step 2 — Evaluate at x = 0 using the zero-exponent rule.

f(0) = 2⁰ = 1

Reason: any non-zero base raised to 0 equals 1; the point (0, 1) is always on y = a^x.

Step 3 — Evaluate at x = 3 by direct multiplication.

f(3) = 2³ = 2 × 2 × 2 = 8

Reason: positive integer exponent means repeated multiplication.

Conclusion. f(−2) = 1/4, f(0) = 1, f(3) = 8.

3. Faded example — fill in the missing steps

Determine whether y = (1/3)^x shows growth or decay, and find its y-intercept. 4 marks

Step 1 — Identify the base.

Base a = ____________

Step 2 — Compare the base to 1.

Since ____ < 1/3 < ____, the function shows ____________ .

Step 3 — Find the y-intercept by setting x = 0.

y = (1/3)⁰ = ____________

Conclusion. y = (1/3)^x shows ____________ with y-intercept (____, ____).

Stuck? Revisit lesson § Worked Example 2.

4. Graduated practice

Show your working in the space provided. Use exact values where possible.

Foundation — direct evaluation (4 questions)

Q	Compute	Answer (exact)
4.1 1	3⁴
4.2 1	5⁰
4.3 1	2⁻³
4.4 1	(1/2)⁴

Standard — classify and evaluate (6 questions)

For each function, state whether it shows growth or decay, and evaluate at the listed x-values.

4.5 f(x) = 4^x; evaluate f(2) and f(−1). State growth/decay. 2 marks

4.6 f(x) = (2/5)^x; evaluate f(0) and f(2). State growth/decay. 2 marks

4.7 f(x) = 10^x; evaluate f(−2), f(0), f(3). 2 marks

4.8 f(x) = (0.5)^x; evaluate f(3). Show that (0.5)³ = 1/8. 2 marks

4.9 A model P(t) = 100 × 2^t gives bacterial population after t hours. Compute P(4). 2 marks

4.10 A car's value follows V(t) = 30 000 × (0.85)^t. Compute V(2) to the nearest dollar. 2 marks

Extension — reason about exponentials (2 questions)

4.11 Without using a calculator, decide which is larger: 2¹⁰ or 10². Show one or two lines of working that justify your choice. 3 marks

4.12 Explain why neither y = 1^x nor y = 0^x nor y = (−2)^x is admitted as an exponential function in the form y = a^x. Give one specific reason for each. 3 marks

Stuck on 4.12? Revisit lesson § Trap 03 — invalid bases.

5. Self-check the easy 3

Tick the first three once you've checked your method works.

For 4.1 (3⁴) I computed 3 × 3 × 3 × 3 = 81, not 3 × 4 = 12.

For 4.3 (2⁻³) I wrote 1/2³ = 1/8 — a positive answer, not −8.

For 4.2 (5⁰) I wrote 1, not 5 or 0.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — Definition

a > 0 and a ≠ 1. (Bases must be positive and not equal to 1.)

Q1.2 — Growth vs decay

If a > 1: exponential growth. If 0 < a < 1: exponential decay.

Q1.3 — y-intercept

(0, 1) — because a⁰ = 1 for any valid base.

Q3 — Faded example y = (1/3)^x

Step 1: a = 1/3.
Step 2: Since 0 < 1/3 < 1, the function shows decay.
Step 3: y = (1/3)⁰ = 1.
Conclusion: y = (1/3)^x shows decay with y-intercept (0, 1).

Q4.1–4.4 — Direct evaluation

4.1: 3⁴ = 81. 4.2: 5⁰ = 1. 4.3: 2⁻³ = 1/8. 4.4: (1/2)⁴ = 1/16.

Q4.5 — f(x) = 4^x

f(2) = 4² = 16. f(−1) = 1/4¹ = 1/4. Base 4 > 1 → growth.

Q4.6 — f(x) = (2/5)^x

f(0) = 1. f(2) = (2/5)² = 4/25 = 0.16. Base 2/5 = 0.4 is between 0 and 1 → decay.

Q4.7 — f(x) = 10^x

f(−2) = 1/100 = 0.01. f(0) = 1. f(3) = 1000.

Q4.8 — f(x) = (0.5)^x

(0.5)³ = 0.5 × 0.5 × 0.5 = 0.125 = 1/8. Note (0.5)³ = (1/2)³ = 1³/2³ = 1/8 confirms this.

Q4.9 — P(t) = 100 × 2^t

P(4) = 100 × 2⁴ = 100 × 16 = 1600 bacteria.

Q4.10 — V(t) = 30 000 × (0.85)^t

V(2) = 30 000 × (0.85)² = 30 000 × 0.7225 = $21 675.

Q4.11 — Which is larger: 2¹⁰ or 10²?

2¹⁰ = 1024. 10² = 100. So 2¹⁰ > 10². Reasoning: 2¹⁰ = (2³)³ × 2 = 8³ × 2 = 512 × 2 = 1024, whereas 10² = 100. This is the typical surprise: exponential growth (variable in the exponent) outpaces polynomial growth (variable in the base) for modest sizes already.

Q4.12 — Why exclude a = 1, 0, −2

a = 1: 1^x = 1 for every x, so the function is the constant y = 1, not an increasing or decreasing exponential — hence we exclude it.
a = 0: 0^x is 0 for x > 0 and is undefined for x ≤ 0 (e.g. 0⁰, 0⁻¹). The function fails to be defined on all real x.
a = −2: A negative base is not defined for many real exponents (e.g. (−2)^1/2 is not real). The resulting "function" is not a function of a real variable in any useful sense.