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Evaluate exponential functions and identify growth vs decay behavior
An exponential function is a function where the variable appears in the exponent. The general form is f(x) = a × b^(kx) + c, where b is the base (b > 0, b ≠ 1), a is the coefficient, k affects the rate, and c is the vertical shift.
For y = a × bˣ: if b > 1 and a > 0, it's exponential growth. If 0 < b < 1 and a > 0, it's exponential decay. The sign of 'a' determines if the function is above or below the asymptote.
The base is the constant that is raised to the variable power. Common bases include e (≈2.71828) for natural exponential functions, 10 for common exponentials, and 2 for binary applications.
The coefficient 'a' scales the function vertically. If |a| > 1, it stretches the graph; if 0 < |a| < 1, it compresses it. A negative 'a' reflects the graph across the horizontal asymptote.
For y = a × b^(kx) + c, the horizontal asymptote is y = c. The function approaches this line but never crosses it (in simple cases). For y = a × bˣ, the asymptote is y = 0.
Yes, if the coefficient 'a' is negative or if there's a negative vertical shift 'c'. However, the base 'b' must always be positive (and not equal to 1) for a true exponential function.
In y = a × b^(kx) + c, the parameter 'k' affects the rate of growth or decay. If |k| > 1, the function changes faster; if 0 < |k| < 1, it changes slower. A negative 'k' with b > 1 creates decay instead of growth.
Exponential functions model population growth, compound interest, radioactive decay, bacteria growth, drug elimination from the body, and many other natural phenomena where the rate of change is proportional to the current value.