Exponential Functions
Grade 10 · Math · Study Guide
Understanding growth, decay, and the properties of exponential functions.
An exponential function has the form f(x) = a · bˣ, where b is the base (b > 0, b ≠ 1) and a is the initial value. The key feature that sets exponential functions apart: the variable is in the exponent, not the base. This creates a dramatically different growth pattern compared to linear or quadratic functions.
When b > 1, the function models exponential growth — the output multiplies by b for every unit increase in x. A classic example is compound interest or a population doubling over time. When 0 < b < 1, the function models exponential decay — the output decreases by a constant percentage each period. Radioactive decay and medication wearing off in the body both follow this pattern.
The graph of an exponential function always passes through (0, a) on the y-axis (since b⁰ = 1). It has a horizontal asymptote — a line the curve approaches but never touches. For f(x) = a · bˣ with a > 0, the asymptote is y = 0. The graph never crosses the x-axis.
The natural base e ≈ 2.71828 shows up constantly in science and finance. The function f(x) = eˣ is its own derivative, which makes it uniquely powerful in calculus. Continuous compound interest uses the formula A = Peʳᵗ, where P is principal, r is annual rate, and t is time in years.
| x | Linear f(x) = 2x | Exponential f(x) = 2ˣ |
|---|---|---|
| 0 | 0 | 1 |
| 2 | 4 | 4 |
| 4 | 8 | 16 |
| 6 | 12 | 64 |
| 10 | 20 | 1024 |
Key terms
- Exponential Function
- f(x) = a·bˣ where b > 0 and b ≠ 1; the variable appears as the exponent.
- Exponential Growth
- When b > 1; the output increases by a constant multiplier each period.
- Exponential Decay
- When 0 < b < 1; the output decreases by a constant percentage each period.
- Horizontal Asymptote
- A horizontal line the graph approaches but never reaches; y = 0 for standard exponential functions.
- Natural Base e
- An irrational constant ≈ 2.71828 that arises naturally in continuous growth models.
Flashcards
- Exponential function
- f(x) = a·bˣ where b > 0, b ≠ 1; variable is the exponent
- Exponential growth
- Base b > 1; function increases rapidly
- Exponential decay
- Base 0 < b < 1; function decreases toward zero
- Y-intercept of a·bˣ
- Always equals a, since b⁰ = 1
- Horizontal asymptote
- y = 0 for standard exponential f(x) = a·bˣ
- Natural base e
- ≈ 2.71828; used in continuous growth/decay models
Practice questions
In f(x) = 3 · 2ˣ, what is the y-intercept?
- 2
- 3
- 6
- 0
Show answer
Answer: 3
f(0) = 3·2⁰ = 3·1 = 3. The y-intercept is always a in f(x) = a·bˣ.
The function f(x) = (1/2)ˣ models exponential decay.
Show answer
Answer: True
Base = 1/2 < 1, so each step halves the output — classic exponential decay.
The horizontal asymptote of f(x) = 5 · 3ˣ is y = ___.
Show answer
Answer: 0
As x→-∞, bˣ→0 but never reaches it. The graph approaches y=0 asymptotically.
Which function represents exponential growth?
- f(x) = 3 · (0.5)ˣ
- f(x) = 2 · (0.8)ˣ
- f(x) = 4 · (1.5)ˣ
- f(x) = 10 · (1)ˣ
Show answer
Answer: f(x) = 4 · (1.5)ˣ
Base 1.5 > 1, so the function grows. Bases 0.5 and 0.8 are less than 1 (decay). Base 1 gives a constant, not exponential.
The graph of an exponential function f(x) = a · bˣ (a > 0, b > 1) ever crosses the x-axis.
Show answer
Answer: False
For a > 0 and b > 0, a·bˣ is always positive. The graph approaches y=0 but never touches or crosses it.
Fun facts
- If you fold a piece of paper in half 42 times (theoretically), the stack would be thick enough to reach the Moon — that's exponential growth in action.
- The value e was discovered by Jacob Bernoulli in 1683 while studying compound interest — he noticed that as compounding frequency increases, growth approaches a specific limit.
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