To find the best fit exponential function for the given data, we need to use the form [tex]\( y = a \cdot b^x \)[/tex]. We'll determine the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] that best fit the data.
Here are the steps broken down:
1. Plot the Data: First, plot the data points [tex]\((x, y)\)[/tex]:
- (0, 11.1)
- (1, 4)
- (2, 1.44)
- (3, 0.52)
- (4, 0.19)
- (5, 0.07)
2. Use a graphing calculator or software to find the best fit: Utilizing a graphing calculator with a curve fitting function such as "Exponential Regression," or software tools to compute, we input the data points and get the best fit parameters [tex]\( a \)[/tex] and [tex]\( b \)[/tex].
3. Interpret the calculator output: Based on the exponential regression analysis, the parameters come out to be:
- [tex]\( a = 11.1 \)[/tex]
- [tex]\( b = 0.36 \)[/tex]
4. Form the equation: With [tex]\( a \)[/tex] and [tex]\( b \)[/tex] known, we write the best fit exponential function as:
[tex]\[
y = 11.1 \cdot 0.36^x
\][/tex]
5. Rounding the parameters: Both [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are accurately rounded to two decimal places in our function.
Therefore, the best fit exponential function for the given data is:
[tex]\[
y = 11.1 \cdot 0.36^x
\][/tex]
This function represents the trend of the provided data points, describing the exponential decay observed in the values of [tex]\( y \)[/tex] as [tex]\( x \)[/tex] increases.
