6.4Introduction to Exponential Functions
Identify, evaluate, and graph exponential functions of the form f(x) = ab^x. Distinguish growth from decay, build tables of values, and write equations from two points.
Why This Matters
Exponential functions model the most dramatic changes in nature — population growth, viral spread, and radioactive decay. They're a major topic in Algebra 2, Precalculus, and AP Calculus BC.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How do exponential functions differ from linear functions, and how do we identify, evaluate, and graph them?
Lesson Overview
An exponential function is a function of the form f(x) = ab^x, where the variable x appears as an exponent. This is fundamentally different from linear functions (where x is multiplied by a constant) and quadratic functions (where x is squared). Exponential functions model situations where a quantity grows or shrinks by a constant factor — not a constant amount — at each step.
Linear vs Exponential Functions
| Feature | Linear | Exponential |
|---|---|---|
| Pattern | Add a constant each step | Multiply by a constant each step |
| Rate of change | Constant (slope) | Variable (grows/shrinks faster) |
| Equation form | y = mx + b | y = ab^x |
| Graph shape | Straight line | Curved (J-shape or decay curve) |
| Example | Save $5 every week | Double your money every year |
| Table pattern | Differences are equal | Ratios are equal |
Exponential Curve: y = 2ˣ
Graph of y = 2^x showing exponential growth
Function Machine: x → 2ˣ
Function machine for y = 2^x
Table → Equation
Table of values for y = 2^x
Anatomy of an Exponential Function
a
Initial Value
The y-intercept; the value of f(x) when x = 0
f(0) = a · b⁰ = a
b
Base (Growth Factor)
Must be positive and not equal to 1
b > 1: growth | 0 < b < 1: decay
x
Exponent (Input)
The independent variable; can be any real number
This is what makes it exponential!
Is It Growth or Decay?
b > 1
GROWTH
Function increases as x increases
e.g. b = 2, 3, 1.5
0 < b < 1
DECAY
Function decreases as x increases
e.g. b = 0.5, 0.8, ¾
Table of Values — f(x) = 2ˣ
| x | f(x) = 2ˣ | Note |
|---|---|---|
| -3 | 1/8 = 0.125 | 2⁻³ = 1/2³ |
| -2 | 1/4 = 0.25 | 2⁻² = 1/2² |
| -1 | 1/2 = 0.5 | 2⁻¹ = 1/2¹ |
| 0 | 1 | 2⁰ = 1 (always!) |
| 1 | 2 | 2¹ = 2 |
| 2 | 4 | 2² = 4 |
| 3 | 8 | 2³ = 8 |
| 4 | 16 | 2⁴ = 16 |
Key observation: Each row's y-value is exactly double the previous row.
Constant ratio = 2. This is the hallmark of an exponential function.
Steps for Graphing an Exponential Function
Identify a and b
Write the function in f(x) = ab^x form. Note whether b > 1 (growth) or 0 < b < 1 (decay).
Find the y-intercept
Substitute x = 0. You always get (0, a). Plot this point first.
Build a table of values
Choose x = −2, −1, 0, 1, 2. Calculate f(x) for each. Plot all five points.
Draw the asymptote
Draw a dashed horizontal line at y = 0. The curve approaches but never touches it.
Connect with a smooth curve
Draw a smooth curve through your points. It should approach the asymptote on one end.
Label key features
Mark the y-intercept, asymptote, and at least two other points. State domain and range.
Growth vs Decay Curves
f(x) = 1 · 2ˣ (b = 2 > 1)
Curve rises steeply — GROWTH
f(x) = 16 · (½)ˣ (b = ½ < 1)
Curve falls steeply — DECAY
Key Properties of f(x) = ab^x
Effect of the Parameter a on f(x) = ab^x
| Value of a | Effect on Graph | Example |
|---|---|---|
| a > 0 | Graph is above the x-axis | f(x) = 3 · 2ˣ |
| a < 0 | Graph is below the x-axis (reflected) | f(x) = −3 · 2ˣ |
| a > 1 | Graph is vertically stretched | f(x) = 5 · 2ˣ |
| 0 < a < 1 | Graph is vertically compressed | f(x) = 0.5 · 2ˣ |
| a = 1 | Standard form — no stretch/compress | f(x) = 1 · 2ˣ = 2ˣ |
How to Identify Exponential from a Table
Linear Table (constant differences)
| x | y | Diff |
|---|---|---|
| 0 | 2 | |
| 1 | 5 | +3 |
| 2 | 8 | +3 |
| 3 | 11 | +3 |
| 4 | 14 | +3 |
Equal differences → Linear
Exponential Table (constant ratios)
| x | y | Ratio |
|---|---|---|
| 0 | 2 | |
| 1 | 6 | ×3 |
| 2 | 18 | ×3 |
| 3 | 54 | ×3 |
| 4 | 162 | ×3 |
Equal ratios → Exponential (b = 3)
Writing an Exponential Equation from Two Points
Use the y-intercept point
If one point is (0, a), then a is the initial value directly. Write f(x) = a · b^x.
Substitute the second point
Plug the second (x, y) pair into f(x) = a · b^x to create an equation with one unknown (b).
Solve for b
Divide both sides by a, then take the appropriate root or use exponent rules to isolate b.
Write the final equation
Substitute both a and b back into f(x) = ab^x. Verify with both original points.
Real-World Exponential Functions
Bacteria Growth
f(t) = 100 · 2^t
Doubles every hour. After 5 hours: 100 · 2⁵ = 3,200 bacteria.
Radioactive Decay
f(t) = 500 · (0.5)^t
Half-life of 1 year. After 3 years: 500 · (0.5)³ = 62.5 grams.
Compound Interest
A = P(1 + r)^t
$1,000 at 5% annually. After 10 years: 1000 · (1.05)¹⁰ ≈ $1,629.
Viral Social Media
f(t) = 1 · 3^t
Each person shares with 3 others. After 5 shares: 3⁵ = 243 people.
Worked Examples
Evaluate f(x) = 3 · 2^x at x = 0, 1, 2, and −1.
f(0) = 3 · 2⁰ = 3 · 1 = 3
f(1) = 3 · 2¹ = 3 · 2 = 6
f(2) = 3 · 2² = 3 · 4 = 12
f(−1) = 3 · 2⁻¹ = 3 · (1/2) = 3/2 = 1.5
Notice: each output is double the previous one (constant ratio = 2).
Evaluate g(x) = 100 · (0.5)^x at x = 0, 1, 2, 3.
g(0) = 100 · (0.5)⁰ = 100 · 1 = 100
g(1) = 100 · (0.5)¹ = 100 · 0.5 = 50
g(2) = 100 · (0.5)² = 100 · 0.25 = 25
g(3) = 100 · (0.5)³ = 100 · 0.125 = 12.5
Each output is half the previous one (constant ratio = 0.5). This is exponential decay.
Is the table below linear or exponential? If exponential, write the equation. Table: x = 0, 1, 2, 3 → y = 5, 15, 45, 135
Check differences: 15−5=10, 45−15=30, 135−45=90. Not constant → not linear.
Check ratios: 15/5=3, 45/15=3, 135/45=3. Constant ratio = 3 → exponential!
Since x=0 gives y=5, the initial value a = 5.
The base b = 3 (the constant ratio).
Equation: f(x) = 5 · 3^x
Verify: f(2) = 5 · 3² = 5 · 9 = 45 ✓
Graph f(x) = 2^x. Identify the y-intercept, asymptote, domain, and range.
a = 1, b = 2. Since b = 2 > 1, this is exponential growth.
Build table: f(−2)=0.25, f(−1)=0.5, f(0)=1, f(1)=2, f(2)=4, f(3)=8.
y-intercept: (0, 1) — plot this first.
Asymptote: y = 0 (draw dashed horizontal line along x-axis).
Plot all points and connect with a smooth curve rising to the right.
Domain: all real numbers (−∞, ∞). Range: y > 0, written as (0, ∞).
Write an exponential equation passing through (0, 6) and (2, 54).
Since one point is (0, 6), the y-intercept gives a = 6.
Write f(x) = 6 · b^x.
Substitute (2, 54): 54 = 6 · b².
Divide both sides by 6: 9 = b².
Take the square root: b = 3 (b must be positive).
Final equation: f(x) = 6 · 3^x.
Verify: f(0) = 6 · 1 = 6 ✓ and f(2) = 6 · 9 = 54 ✓
Guided Practice
Guided Practice Video: Introduction to Exponential Functions
Watch the guided practice walkthrough for introduction to exponential functions, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Evaluate f(x) = 4 · 3^x at x = 0, 1, 2, and −1.
Hint: Substitute each x-value. Remember 3⁰ = 1 and 3⁻¹ = 1/3.
Is f(x) = 6 · (0.4)^x growth or decay? Explain.
Hint: Look at the base b = 0.4. Is it greater than 1 or between 0 and 1?
Build a table of values for f(x) = 2 · 5^x using x = −1, 0, 1, 2.
Hint: Calculate 5^(−1) = 1/5 = 0.2, then multiply by 2 each time.
Identify the y-intercept and asymptote of f(x) = 7 · (0.3)^x.
Hint: y-intercept: substitute x = 0. Asymptote: what value does the function approach but never reach?
A table shows x = 0, 1, 2, 3 → y = 3, 12, 48, 192. Write the exponential equation.
Hint: Find the constant ratio (divide consecutive y-values). The y-intercept gives a.
Key Vocabulary
Exponential Function
A function of the form f(x) = ab^x where b > 0 and b ≠ 1; the variable is in the exponent.
Initial Value (a)
The value of the function when x = 0; equals the y-intercept. f(0) = a.
Base (b)
The constant factor by which the output is multiplied for each unit increase in x. Must satisfy b > 0 and b ≠ 1.
Exponential Growth
When b > 1, the function increases as x increases. The graph rises steeply to the right.
Exponential Decay
When 0 < b < 1, the function decreases as x increases. The graph falls and approaches zero.
Horizontal Asymptote
A horizontal line that the graph approaches but never crosses. For f(x) = ab^x, the asymptote is y = 0.
Growth Factor
The base b in an exponential growth function. Represents the multiplier per unit of x.
Decay Factor
The base b in an exponential decay function (0 < b < 1). Represents the fraction remaining per unit of x.
Interactive Practice — 5 Questions
Which function is exponential?
In f(x) = 5 · 3^x, what is the y-intercept?
Which base produces exponential decay?
Evaluate f(x) = 2 · 4^x at x = 3.
What is the horizontal asymptote of f(x) = 6 · 2^x?
Independent Practice
Independent Practice
Evaluate f(x) = 2 · 4^x at x = 0, 1, 2, 3. Then evaluate g(x) = 500 · (0.8)^x at x = 0, 1, 2, 5. Show all work.
Identify a and b in each function. State whether it is growth or decay: (a) f(x) = 0.5 · 6^x (b) h(x) = 1000 · (0.9)^x (c) k(x) = 3 · (1/4)^x.
Build a table for f(x) = 3^x using x = −2, −1, 0, 1, 2. Then state the domain, range, y-intercept, and asymptote.
Write an exponential equation through (0, 4) and (2, 36). Also write one through (0, 100) and (3, 12.5). Show all steps.
A population of 500 triples every year. Write a function and find the population after 4 years. A radioactive sample of 1,000 g has a half-life of 1 year — write a function and find the amount after 5 years.
Common Mistakes
Confusing the initial value (a) with the base (b) — e.g., writing f(x) = b · aˣ.
The standard form is f(x) = a · bˣ where a is the initial value (y-intercept) and b is the growth/decay factor.
Thinking b > 1 means decay and 0 < b < 1 means growth.
b > 1 means exponential growth. 0 < b < 1 means exponential decay.
Evaluating b⁰ as 0 instead of 1 — e.g., f(0) = 3 · 2⁰ = 0.
Any nonzero base to the zero power equals 1: f(0) = 3 · 2⁰ = 3 · 1 = 3. The y-intercept is always a.
Confusing exponential functions with linear functions when the table shows a constant ratio.
Linear functions have a constant difference between outputs. Exponential functions have a constant ratio (multiplier) between outputs.
Math Tips
The variable must be in the exponent: f(x) = 2^x is exponential; f(x) = 2x is linear.
The y-intercept is always (0, a): substitute x = 0 to confirm — b⁰ = 1, so f(0) = a.
b > 1 means growth (curve rises); 0 < b < 1 means decay (curve falls toward zero).
Constant ratio in a table = exponential. Constant difference in a table = linear.
The horizontal asymptote is always y = 0 — the graph approaches but never crosses the x-axis.