2.2 Modeling data sets with exponential functions

• In the exponential function $y =$$ab^{x}$, the variable a</latex> represents the initial value
• What type of data change is best modeled by an exponential function?

  Fixed percentage change
• The percent change between consecutive data points in an exponential model is approximately constant.

  True
• What is the general form of an exponential function?

  $y =$$ab^{x}$
• What happens to an exponential function when $0 < b < 1$?


  It decays
• In the example $y =$$10(0.5)^{x}$, the function decays because b < 1</latex>.


  True
• What is the formula to find $b$ using the point-slope method?


  $b =$$\left( \frac{y_{2}}{y_{1}} \right)^{\frac{1}{x_{2} - x_{1}}}$
• What are the data points used to define the exponential function?

  $(x_{1}, y_{1})$, $(x_{2}, y_{2})$
• If $b > 1$ in an exponential function, the function grows.


  True
• What are the values of $a$ and $b$ in the exponential function $y =$$10(0.5)^{x}$?


  $a =$$10, b =$$0.5$
• If the growth factor $b$ in an exponential function is 1.5, the function is increasing.


  True
• The rate of change in an exponential model is not constant, but rather increases or decreases
• In exponential models, data points form a curved line when plotted.
• In an exponential function, $a$ represents the initial value when $x =$$0$.
• Match the variable in $y =$$ab^{x}$ with its meaning:


  $a$ ↔️ Initial value

  $b$ ↔️ Growth/decay factor
• What is the percent change in the example exponential data?

  50%
• To solve for $a$, you need $b$ and one data point.
• The growth/decay factor $b$ is calculated using the formula b = \left( \frac{y_{2}}{y_{1}} \right)^{\frac{1}{x_{2} - x_{1}}}</latex>, which requires two data points.
• The initial value $a$ for the data points $(0, 10)$ and $(2, 2.5)$ is 10.
• The growth/decay factor $b$ is always positive for exponential functions.


  True
• What is the general form of an exponential function?

  $y =$$ab^{x}$
• What are the values of $a$ and b</latex> in the exponential function $y =$$3(2)^{x}$?


  $a =$$3, b =$$2$
• If $0 < b < 1$ in an exponential function, the function decays
• What is the initial value of the exponential function $y =$$3(2)^{x}$?


  $a =$$3$
• What type of rate of change is characteristic of exponential models?

  Non-constant
• What is an example of a scenario suitable for exponential modeling?

  Population growth
• If $b > 1$ in an exponential function, the function grows.


  True
• In the example $y =$$3(2)^{x}$, the growth factor is 2.
• Steps to find an exponential function using the point-slope method:

  1️⃣ Choose two data points

  2️⃣ Determine $b$ using the formula

  3️⃣ Solve for $a$ in $y =$$ab^{x}$
• The exponential function found using the data points $(0, 10)$ and $(2, 2.5)$ is y = 10(0.5)^{x}</latex>.


  True
• The initial value $a$ is found using the growth/decay factor $b$ and one data point.


  True
• What is the exponential function for the data points $(0, 10)$ and $(2, 2.5)$?


  $y =$$10(0.5)^{x}$
• To verify an exponential model, substitute the $x$ values from the data into the model to obtain calculated $y$ values, then compare these values with the original data.
• What should you look for to identify if data is suitable for exponential modeling?

  Non-constant rate of change
• Verifying the exponential model involves comparing calculated $y$ values with original data points.


  True
• Data suitable for exponential modeling exhibits a curved line when graphed.
• The growth/decay factor b in exponential modeling is calculated using the formula (y2/y1)^(1/(x2-x1)).
• What is the y-value when x = 0 in the sample data set?

  10
• What is the exponential function that models the sample data set?

  $y =$$10(1.5)^{x}$
• What is the calculated y value when x = 0 using the exponential model for the sample data?

  10