Explanation
Definition
Exponential functions are functions in which the variable appears in the exponent. They have the form f(x) = a \cdot b^x. Here, a is the initial value and b is the growth factor. If b > 1, the function will grow, whereas if 0 < b < 1, it decreases.
f(x) = a \cdot b^x
Procedure
Scheme
Formulate the function term
f(x) = a \cdot b^x
Determine the initial value
f(0) = a
Determine the growth factor
b = \frac{f(x+1)}{f(x)}
Analyze properties
If b > 1, the function grows; at 0 < b < 1, it decreases.
Misunderstandings
Common Errors
- ★It is often forgotten to treat the exponent as a variable rather than a multiplier.
- ★It is often overlooked that f(0) = a denotes the initial value.
- ★Often the growth factor b with the initial valuea is confused
Problems
1 / 3
Example
f(x) = 3 \cdot 2^x
2 / 3
Example
f(0) = 5, f(1) = 10
3 / 3
Example
f(1) = 5,\; f(3) = 20
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