To find the equation of the linear function that fits the given data points, we need to follow a structured approach:
### Step 1: List the provided data points
We have the following data points:
[tex]\[
\begin{array}{|c|c|}
\hline x & y \\
\hline -5 & 14 \\
\hline -2 & 11 \\
\hline 1 & 8 \\
\hline 4 & 5 \\
\hline
\end{array}
\][/tex]
### Step 2: Identify the potential linear equations
The possible linear functions given are:
1. [tex]\( y = -x + 9 \)[/tex]
2. [tex]\( y = -x + 13 \)[/tex]
3. [tex]\( y = x + 13 \)[/tex]
4. [tex]\( y = x + 9 \)[/tex]
### Step 3: Verify each equation against the data points
We'll substitute each [tex]\( x \)[/tex] value from the table into the candidate linear functions to see if the resulting [tex]\( y \)[/tex] values match those in the table.
#### Equation 1: [tex]\( y = -x + 9 \)[/tex]
Substitute [tex]\( x \)[/tex] and compute [tex]\( y \)[/tex]:
- For [tex]\( x = -5 \)[/tex]: [tex]\( y = -(-5) + 9 = 5 + 9 = 14 \)[/tex]
- For [tex]\( x = -2 \)[/tex]: [tex]\( y = -(-2) + 9 = 2 + 9 = 11 \)[/tex]
- For [tex]\( x = 1 \)[/tex]: [tex]\( y = -(1) + 9 = -1 + 9 = 8 \)[/tex]
- For [tex]\( x = 4 \)[/tex]: [tex]\( y = -(4) + 9 = -4 + 9 = 5 \)[/tex]
Since all computed [tex]\( y \)[/tex] values match the [tex]\( y \)[/tex] values in the table, the equation [tex]\( y = -x + 9 \)[/tex] is correct.
#### Equation 2: [tex]\( y = -x + 13 \)[/tex]
Substitute [tex]\( x \)[/tex] and compute [tex]\( y \)[/tex]:
- For [tex]\( x = -5 \)[/tex]: [tex]\( y = -(-5) + 13 = 5 + 13 = 18 \)[/tex]
- For [tex]\( x = -2 \)[/tex]: [tex]\( y = -(-2) + 13 = 2 + 13 = 15 \)[/tex]
- For [tex]\( x = 1 \)[/tex]: [tex]\( y = -(1) + 13 = -1 + 13 = 12 \)[/tex]
- For [tex]\( x = 4 \)[/tex]: [tex]\( y = -(4) + 13 = -4 + 13 = 9 \)[/tex]
The computed [tex]\( y \)[/tex] values do not match the [tex]\( y \)[/tex] values in the table.
#### Equation 3: [tex]\( y = x + 13 \)[/tex]
Substitute [tex]\( x \)[/tex] and compute [tex]\( y \)[/tex]:
- For [tex]\( x = -5 \)[/tex]: [tex]\( y = -5 + 13 = 8 \)[/tex]
- For [tex]\( x = -2 \)[/tex]: [tex]\( y = -2 + 13 = 11 \)[/tex]
- For [tex]\( x = 1 \)[/tex]: [tex]\( y = 1 + 13 = 14 \)[/tex]
- For [tex]\( x = 4 \)[/tex]: [tex]\( y = 4 + 13 = 17 \)[/tex]
The computed [tex]\( y \)[/tex] values do not match the [tex]\( y \)[/tex] values in the table (only some of them match).
#### Equation 4: [tex]\( y = x + 9 \)[/tex]
Substitute [tex]\( x \)[/tex] and compute [tex]\( y \)[/tex]:
- For [tex]\( x = -5 \)[/tex]: [tex]\( y = -5 + 9 = 4 \)[/tex]
- For [tex]\( x = -2 \)[/tex]: [tex]\( y = -2 + 9 = 7 \)[/tex]
- For [tex]\( x = 1 \)[/tex]: [tex]\( y = 1 + 9 = 10 \)[/tex]
- For [tex]\( x = 4 \)[/tex]: [tex]\( y = 4 + 9 = 13 \)[/tex]
The computed [tex]\( y \)[/tex] values do not match the [tex]\( y \)[/tex] values in the table.
### Conclusion
After verifying all candidate equations, we find that the equation [tex]\( y = -x + 9 \)[/tex] accurately represents the data points provided in the table. Therefore, the equation of the linear function is:
[tex]\[
y = -x + 9
\][/tex]
