To find the best function that models the given data points [tex]\((x, y)\)[/tex], we will look at the possibility of a linear function of the form [tex]\(y = mx + b\)[/tex].
Given data points:
[tex]\[
\begin{array}{|c|c|}
\hline x & y \\
\hline 0 & -5 \\
\hline 1 & -11 \\
\hline 2 & -17 \\
\hline 3 & -23 \\
\hline 4 & -29 \\
\hline
\end{array}
\][/tex]
We note the [tex]\(y\)[/tex] values decrease as [tex]\(x\)[/tex] increases, suggesting a possible linear relationship. A linear function has the form:
[tex]\[ y = mx + b \][/tex]
Step 1: Calculate the slope [tex]\(m\)[/tex]
[tex]\[
m = \frac{\Delta y}{\Delta x}
\][/tex]
Choose two points to find the slope, for instance, using [tex]\((0, -5)\)[/tex] and [tex]\((4, -29)\)[/tex]:
[tex]\[
m = \frac{-29 - (-5)}{4 - 0} = \frac{-29 + 5}{4} = \frac{-24}{4} = -6
\][/tex]
Step 2: Calculate the intercept [tex]\(b\)[/tex]
To find [tex]\(b\)[/tex], we can use the equation of the line and one of the points, say [tex]\((0, -5)\)[/tex]:
[tex]\[
-5 = -6(0) + b \implies b = -5
\][/tex]
Thus, the linear function that models the data is:
[tex]\[
y = -6x - 5
\][/tex]
Conclusion:
The best function that fits the given data points is a linear function:
[tex]\[
y = -6x - 5
\][/tex]
