To analyze the given table of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values, follow these steps:
1. Understand the Data:
- We have pairs of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values as follows:
- When [tex]\( x = -3 \)[/tex], [tex]\( y = 2 \)[/tex]
- When [tex]\( x = -2 \)[/tex], [tex]\( y = 0 \)[/tex]
- When [tex]\( x = -1 \)[/tex], [tex]\( y = 0 \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 2 \)[/tex]
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 6 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 12 \)[/tex]
- When [tex]\( x = 3 \)[/tex], [tex]\( y = 20 \)[/tex]
2. Identify Patterns:
- Look at the changes in [tex]\( y \)[/tex] for each successive value of [tex]\( x \)[/tex].
3. Calculate Differences:
- First differences:
- [tex]\( y(-2) - y(-3) = 0 - 2 = -2 \)[/tex]
- [tex]\( y(-1) - y(-2) = 0 - 0 = 0 \)[/tex]
- [tex]\( y(0) - y(-1) = 2 - 0 = 2 \)[/tex]
- [tex]\( y(1) - y(0) = 6 - 2 = 4 \)[/tex]
- [tex]\( y(2) - y(1) = 12 - 6 = 6 \)[/tex]
- [tex]\( y(3) - y(2) = 20 - 12 = 8 \)[/tex]
- Second differences (checking for a constant second difference, which indicates a quadratic relationship):
- [tex]\( 0 - (-2) = 2 \)[/tex]
- [tex]\( 2 - 0 = 2 \)[/tex]
- [tex]\( 4 - 2 = 2 \)[/tex]
- [tex]\( 6 - 4 = 2 \)[/tex]
- [tex]\( 8 - 6 = 2 \)[/tex]
4. Conclusion:
- The constant second difference suggests [tex]\( y \)[/tex] could be represented as a quadratic function of [tex]\( x \)[/tex].
Understanding the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] based on the observations can be summarized thus:
- As [tex]\( x \)[/tex] transitions from [tex]\(-3\)[/tex] to [tex]\(3\)[/tex], the corresponding [tex]\( y \)[/tex] values evolve in accordance with a quadratic pattern, as indicated by the constant second differences.
