Plot the following sets of data and write the best-fit exponential function using a graphing calculator.

\begin{tabular}{|rr|}
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
0 & -0.7 \\
1 & -1.06 \\
2 & -1.62 \\
3 & -2.46 \\
4 & -3.74 \\
5 & -5.68 \\
\end{tabular}

Write the best-fit exponential function in the form [tex]y = ab^x[/tex]. Round to 2 decimal places.



Answer :

Certainly! Let's derive the best fit exponential function [tex]\( y = a b^x \)[/tex] from the given data:

The data points provided are:

[tex]\[ \begin{array}{c|c} x & y \\ \hline 0 & -0.7 \\ 1 & -1.06 \\ 2 & -1.62 \\ 3 & -2.46 \\ 4 & -3.74 \\ 5 & -5.68 \\ \end{array} \][/tex]

To find the best fit exponential function [tex]\( y = a b^x \)[/tex], we perform exponential regression, which allows us to fit a curve of the form [tex]\( y = a b^x \)[/tex] to the data points. After carrying out the exponential regression process, we determine the parameters [tex]\( a \)[/tex] and [tex]\( b \)[/tex].

The calculated parameters (rounded to two decimal places) are:
[tex]\[ a = -0.70 \][/tex]
[tex]\[ b = 1.52 \][/tex]

Therefore, the best fit exponential function for the given data is:
[tex]\[ y = -0.70 \cdot 1.52^x \][/tex]

So, the equation of the curve that best fits the given data points is:
[tex]\[ y = -0.70 \cdot 1.52^x \][/tex]