Certainly! Let's plot the given sets of data and write the best fit exponential function in the form [tex]\( y = a \cdot b^x \)[/tex]. Here’s how to approach it step-by-step.
### Step-by-Step Solution:
1. Organize the Data:
The given data points are:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & -0.7 \\
1 & -1.06 \\
2 & -1.62 \\
3 & -2.46 \\
4 & -3.74 \\
5 & -5.68 \\
\hline
\end{array}
\][/tex]
2. Interpret the Data:
These points suggest an exponential decay, as the value of [tex]\( y \)[/tex] is becoming more negative as [tex]\( x \)[/tex] increases. Typically, an exponential function that models such data can be written as:
[tex]\[
y = a \cdot b^x
\][/tex]
where:
- [tex]\( a \)[/tex] is the initial amount (the value when [tex]\( x = 0 \)[/tex]),
- [tex]\( b \)[/tex] is the base of the exponential which shows the rate of growth or decay.
3. Plot the Data Points:
When you input these points into a graphing calculator or plotting tool, you will get a scatter plot showing these points on a coordinate plane.
4. Fit an Exponential Function:
Use a graphing calculator or software to fit an exponential curve to these data points. The tool uses methods such as least squares fitting to find the best fit exponential function.
5. Determine the Coefficients [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
After fitting the curve, the calculator provides the coefficients for the best fit. According to our results:
[tex]\[
a = -0.7, \quad b = 1.52
\][/tex]
We round these to two decimal places if necessary (they are already in this form).
6. Write the Exponential Function:
Substitute [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the general form of the exponential function:
[tex]\[
y = -0.7 \cdot (1.52)^x
\][/tex]
### Final Best Fit Exponential Function:
[tex]\[
y = -0.7 \cdot 1.52^x
\][/tex]
This is the exponential function that best fits your data points, rounded to two decimal places.
