Exponential Functions

Grade 9 · Math · Study Guide

Understand exponential growth and decay, graph exponential functions, and solve exponential equations.

An exponential function has the form f(x) = a · bˣ, where a is the initial value (y-intercept when x = 0), b is the base (a positive number not equal to 1), and x is the exponent. When b > 1, the function models exponential growth. When 0 < b < 1, the function models exponential decay.

The graph of f(x) = bˣ always passes through (0, 1) because b⁰ = 1. As x increases, growth functions rise rapidly while decay functions approach but never reach zero — the x-axis is a horizontal asymptote.

Common applications include population growth, compound interest, and radioactive decay. The compound interest formula A = P(1 + r/n)^(nt) is an exponential function where P is principal, r is rate, n is compounding periods per year, and t is time.

To solve simple exponential equations, rewrite both sides with the same base and set the exponents equal. For example, 2ˣ = 32 becomes 2ˣ = 2⁵, so x = 5. Transformations such as y = a · bˣ + k shift the graph vertically by k, changing the horizontal asymptote to y = k.

Key terms

Exponential function: A function of the form f(x) = a · bˣ where b > 0 and b ≠ 1
Exponential growth: An exponential function where b > 1, causing the output to increase rapidly
Exponential decay: An exponential function where 0 < b < 1, causing the output to decrease toward zero
Horizontal asymptote: A horizontal line that a graph approaches but never touches, e.g., y = 0 for basic exponential functions
Initial value: The value of a function when x = 0; for f(x) = a · bˣ, the initial value is a
Common base method: A technique for solving exponential equations by rewriting both sides with the same base and equating exponents

Flashcards

f(x) = 2ˣ at x = 0: f(0) = 2⁰ = 1
f(x) = 5 · (1/2)ˣ — growth or decay?: Decay, because the base 1/2 is between 0 and 1
Horizontal asymptote of f(x) = 3ˣ: y = 0 (the x-axis)
Solve 2ˣ = 64: x = 6, because 2⁶ = 64
Solve 5ˣ = 1: x = 0, because any base raised to 0 equals 1
Growth factor vs decay factor: Growth: b > 1 (e.g., 1.03 means 3% growth). Decay: 0 < b < 1 (e.g., 0.97 means 3% decay)
f(x) = 10 · (1.2)ˣ — what is the growth rate?: 20% growth per period (b = 1.2 = 1 + 0.20)
Domain and range of f(x) = bˣ (b > 0, b ≠ 1): Domain: all real numbers; Range: y > 0 (all positive reals)

Questions

Which function represents exponential growth?
1. f(x) = 3 · (0.5)ˣ
2. f(x) = 2 · (1.5)ˣ
3. f(x) = 5 · (0.8)ˣ
4. f(x) = (0.25)ˣ
Show answer
Answer: f(x) = 2 · (1.5)ˣ
Exponential growth requires b > 1. Only f(x) = 2 · (1.5)ˣ has a base greater than 1.
For f(x) = 4 · 2ˣ, the y-intercept is __.
Show answer
Answer: 4
The y-intercept occurs at x = 0. f(0) = 4 · 2⁰ = 4 · 1 = 4.
Solve: 3ˣ = 81
1. x = 3
2. x = 4
3. x = 27
4. x = 2
Show answer
Answer: x = 4
Express 81 as a power of 3: 81 = 3⁴. So 3ˣ = 3⁴ means x = 4.
The graph of f(x) = 2ˣ crosses the x-axis.
Show answer
Answer: False
f(x) = 2ˣ is always positive (the range is y > 0). The graph approaches y = 0 but never touches it, so it never crosses the x-axis.
A population of 500 bacteria doubles every hour. Which function models the population after t hours?
1. P(t) = 500 + 2t
2. P(t) = 500 · 2ᵗ
3. P(t) = 2 · 500ᵗ
4. P(t) = 500t²
Show answer
Answer: P(t) = 500 · 2ᵗ
Starting with 500 and doubling each hour means multiplying by 2 for each hour elapsed: P(t) = 500 · 2ᵗ.

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