Let's analyze the table of values given, which lists pairs of [tex]\(x\)[/tex] and corresponding [tex]\(y\)[/tex] values:
[tex]\[
\begin{array}{c|c}
x & y \\
\hline
-2 & 6 \\
-1 & 2 \\
0 & 2 \\
1 & 6 \\
2 & 14 \\
\end{array}
\][/tex]
### Step-by-Step Analysis
1. Identify Patterns in the Data:
- For [tex]\( x = -2 \)[/tex], [tex]\( y = 6 \)[/tex]
- For [tex]\( x = -1 \)[/tex], [tex]\( y = 2 \)[/tex]
- For [tex]\( x = 0 \)[/tex], [tex]\( y = 2 \)[/tex]
- For [tex]\( x = 1 \)[/tex], [tex]\( y = 6 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( y = 14 \)[/tex]
2. Examining Differences to Determine Relationships:
- Calculate the first differences of [tex]\(y\)[/tex]:
[tex]\[
\Delta y_1 = y(-1) - y(-2) = 2 - 6 = -4
\][/tex]
[tex]\[
\Delta y_2 = y(0) - y(-1) = 2 - 2 = 0
\][/tex]
[tex]\[
\Delta y_3 = y(1) - y(0) = 6 - 2 = 4
\][/tex]
[tex]\[
\Delta y_4 = y(2) - y(1) = 14 - 6 = 8
\][/tex]
3. Examine Second Differences:
- Calculate the second differences of [tex]\(y\)[/tex] to see if there is a constant change:
[tex]\[
\Delta^2 y_1 = \Delta y_2 - \Delta y_1 = 0 - (-4) = 4
\][/tex]
[tex]\[
\Delta^2 y_2 = \Delta y_3 - \Delta y_2 = 4 - 0 = 4
\][/tex]
[tex]\[
\Delta^2 y_3 = \Delta y_4 - \Delta y_3 = 8 - 4 = 4
\][/tex]
### Conclusion
The second differences are constant and equal to 4, indicating that [tex]\(y\)[/tex] is a quadratic function of [tex]\(x\)[/tex]. The form of such a function can be [tex]\(y = ax^2 + bx + c\)[/tex]. Finding the exact coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] would typically give us the complete formula, but from the data alone, we have effectively understood how the [tex]\(y\)[/tex] values change with respect to [tex]\(x\)[/tex] values.
Thus our data set can be represented in terms of the patterns we discussed:
- [tex]\( x \)[/tex] values: [-2, -1, 0, 1, 2]
- [tex]\( y \)[/tex] values: [6, 2, 2, 6, 14]
