Exponential Functions and Growth Models
Master key concepts, formulas, and properties of exponential functions and their real-world applications.
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Questions Covered in This Set
10 cards to master
What is the general form of an exponential function?
f(x) = a · b^x, where a is the initial value, b is the base (growth/decay factor), and x is the exponent
What determines whether an exponential function models growth or decay?
The base b: if b > 1 it's growth, if 0 < b < 1 it's decay
What is the y-intercept of any exponential function f(x) = a · b^x?
The point (0, a), because b^0 = 1, so f(0) = a
What is the horizontal asymptote of f(x) = a · b^x (for positive a)?
The line y = 0 (the x-axis); the graph approaches but never touches it
What is the compound interest formula?
A(t) = P(1 + r/n)^(nt), where P is principal, r is annual rate, n is compounds per year, and t is time in years
What is the domain and range of f(x) = a · b^x (positive a)?
Domain: all real numbers (-∞, ∞); Range: (0, ∞)
How do you write an exponential decay function using decay rate r?
f(x) = a · (1 - r)^x, where r is the decay rate between 0 and 1
In population growth P(t) = 100 · 2^t, what does the 2 represent?
The growth factor; the population doubles with each unit increase in t
What is the half-life formula for radioactive decay?
N(t) = N₀ · (1/2)^(t/h), where N₀ is initial amount, t is time, and h is half-life
How does f(x) = a · b^(x-h) + k transform the basic exponential function?
Shifts h units horizontally, k units vertically, and moves the horizontal asymptote to y = k
