To find a model that fits the given data, we first consider different types of functions: linear, quadratic, and exponential.
Given:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-10 & 89 \\
\hline
-9 & 80 \\
\hline
-8 & 71 \\
\hline
-7 & 62 \\
\hline
-6 & 53 \\
\hline
\end{array}
\][/tex]
### Linear Model:
For a linear model [tex]\( y = mx + b \)[/tex], we fit the data points and determine the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex].
The linear model that fits the data is:
[tex]\[
y = -9x - 1
\][/tex]
### Quadratic Model:
For a quadratic model [tex]\( y = ax^2 + bx + c \)[/tex], we fit the data points and determine the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex].
The quadratic model that fits the data is:
[tex]\[
y = -2.29168713 \times 10^{-15}x^2 - 9x - 1
\][/tex]
### Sum of Squared Residuals (SSR):
To determine the best fit, we compute the sum of squared residuals for each model.
- For the linear model: The sum of squared residuals is approximately [tex]\( 8.58 \times 10^{-28} \)[/tex]
- For the quadratic model: The sum of squared residuals is approximately [tex]\( 1.06 \times 10^{-27} \)[/tex]
### Best Fit Model:
The model with the lowest sum of squared residuals is considered the best fit. Here, the linear model has the lowest SSR.
Therefore, the function that best models the data is:
[tex]\[
y = -9x - 1
\][/tex]
This linear equation [tex]\( y = -9x - 1 \)[/tex] serves as the best fit for the given data.
