To find a function that models the given data points, we need to determine the best-fit equation that can describe the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. For the data provided, we're looking for a linear function of the form [tex]\(y = mx + b\)[/tex].
Here is how we can find the coefficients [tex]\(m\)[/tex] (the slope) and [tex]\(b\)[/tex] (the y-intercept):
### Step-by-Step Solution:
1. Identify the given data points:
The provided data points are:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
2 & 2 \\
\hline
3 & 0 \\
\hline
4 & -2 \\
\hline
5 & -4 \\
\hline
6 & -6 \\
\hline
\end{array}
\][/tex]
2. Check the nature of the relationship:
Observe that as [tex]\(x\)[/tex] increases, [tex]\(y\)[/tex] decreases linearly. This suggests a linear relationship.
3. Formulate the linear equation:
A linear relationship can be expressed as:
[tex]\[
y = mx + b
\][/tex]
4. Determine the slope [tex]\(m\)[/tex] and y-intercept [tex]\(b\)[/tex]:
When we fit the data to a linear equation, we find:
[tex]\[
m = -2 \quad \text{and} \quad b = 6
\][/tex]
5. Write the final equation:
Substituting [tex]\(m\)[/tex] and [tex]\(b\)[/tex] into the linear equation, we get:
[tex]\[
y = -2x + 6
\][/tex]
Therefore, the linear function that models the given data is:
[tex]\[
y = -2x + 6
\][/tex]
