Integrated Math 1 / Unit 5: Exponential Functions and Geometric Sequences
5.1 Exponential Functions
Notes
An exponential function has the form $y = a \cdot b^x$ where $a$ is the initial value (at $x = 0$) and $b > 0$, $b \neq 1$, is the growth or decay factor.
- The variable is in the exponent, so equal steps in $x$ multiply $y$ by the same factor.
- Linear functions add a constant; exponential functions multiply by a constant.
- Example: $f(x) = 5 \cdot 2^x$, so $f(3) = 5 \cdot 8 = 40$.
- Order matters: evaluate the power first, then multiply by $a$.
Practice
Which function is exponential?
Only $3 \cdot 2^x$ has the variable in the exponent.
If $f(x) = 5 \cdot 2^x$, what is $f(3)$?
$2^3 = 8$, and $5 \cdot 8 = 40$.
In $y = 7(1.5)^x$, what are the initial value and the growth factor?
The coefficient is the initial value and the base is the growth factor.
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