๐ Topic Summary
Exponential functions model situations where a quantity increases or decreases by a constant percentage over equal intervals. To write an exponential function from two data points, you'll use the form $y = ab^x$, where $a$ is the initial value and $b$ is the growth/decay factor. The key is to substitute the coordinates of your data points into the equation and solve for $a$ and $b$. Remember that $b > 0$ and $b \neq 1$.
The general formula for finding the exponential equation when given two points $(x_1, y_1)$ and $(x_2, y_2)$ is to solve the system:
$y_1 = ab^{x_1}$ and $y_2 = ab^{x_2}$. Dividing the two equations eliminates $a$, making it easier to solve for $b$.
๐ง Part A: Vocabulary
- ๐ Exponential Growth: The rate of growth becomes rapid in proportion to the growing total number or size.
- ๐ Exponential Decay: A decrease in a quantity according to the law $N(t) = N_0e^{-kt}$
- ๐ Data Point: A single item on a graph or in a set of data.
- ๐งฎ Growth Factor: The factor by which a quantity multiplies itself over a period. Represented by $b$ in $y=ab^x$.
- ๐งช Initial Value: The starting value of a function. Represented by $a$ in $y=ab^x$.
โ๏ธ Part B: Fill in the Blanks
An exponential function can be written in the form $y = ab^x$, where 'a' represents the ________ and 'b' represents the ________. If 'b' is greater than 1, the function shows ________. If 'b' is between 0 and 1, the function shows ________. To determine these values from data, we substitute the $x$ and $y$ values of our data ________ into the equation and solve for the missing variables.
๐ค Part C: Critical Thinking
Describe a real-world scenario, other than population growth or radioactive decay, that can be modeled by an exponential function. Explain how you would collect data to determine the specific parameters of the function.