Answer :
To determine which tables represent quadratic relationships, we need to analyze the second differences of the [tex]\( y \)[/tex]-values in each table. A table represents a quadratic relationship if the second differences of the [tex]\( y \)[/tex]-values are constant.
Let's examine each table step-by-step.
### Table 1
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & 1 & 2 & 4 & 8 \\ \hline \end{array} \][/tex]
First differences:
[tex]\[ 2 - 1 = 1, \quad 4 - 2 = 2, \quad 8 - 4 = 4 \][/tex]
Second differences:
[tex]\[ 2 - 1 = 1, \quad 4 - 2 = 2 \][/tex]
The second differences are not constant (not all equal), so this table does not represent a quadratic relationship.
### Table 2
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & -2 & -4 & -8 & -16 \\ \hline \end{array} \][/tex]
First differences:
[tex]\[ -4 - (-2) = -2, \quad -8 - (-4) = -4, \quad -16 - (-8) = -8 \][/tex]
Second differences:
[tex]\[ -4 - (-2) = -2, \quad -8 - (-4) = -4 \][/tex]
The second differences are not constant, so this table does not represent a quadratic relationship.
### Table 3
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & 3 & 4 & 5 & 6 \\ \hline \end{array} \][/tex]
First differences:
[tex]\[ 4 - 3 = 1, \quad 5 - 4 = 1, \quad 6 - 5 = 1 \][/tex]
Second differences:
[tex]\[ 1 - 1 = 0, \quad 1 - 1 = 0 \][/tex]
The second differences are constant (all zero), indicating that this table represents a quadratic relationship.
### Table 4
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & 4 & -4 & -4 & 4 \\ \hline \end{array} \][/tex]
First differences:
[tex]\[ -4 - 4 = -8, \quad -4 - (-4) = 0, \quad 4 - (-4) = 8 \][/tex]
Second differences:
[tex]\[ 0 - (-8) = 8, \quad 8 - 0 = 8 \][/tex]
The second differences are constant (all equal to 8), indicating that this table represents a quadratic relationship.
### Table 5
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & -2 & 0 & 2 & 4 \\ \hline \end{array} \][/tex]
First differences:
[tex]\[ 0 - (-2) = 2, \quad 2 - 0 = 2, \quad 4 - 2 = 2 \][/tex]
Second differences:
[tex]\[ 2 - 2 = 0, \quad 2 - 2 = 0 \][/tex]
The second differences are constant (all zero), indicating that this table represents a quadratic relationship.
### Table 6
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & -4 & -8 & -10 & -10 \\ \hline \end{array} \][/tex]
First differences:
[tex]\[ -8 - (-4) = -4, \quad -10 - (-8) = -2, \quad -10 - (-10) = 0 \][/tex]
Second differences:
[tex]\[ -2 - (-4) = 2, \quad 0 - (-2) = 2 \][/tex]
The second differences are constant (all equal to 2), indicating that this table represents a quadratic relationship.
Based on this analysis, the tables representing quadratic relationships are:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & 3 & 4 & 5 & 6 \\ \hline \end{array} \][/tex]
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & 4 & -4 & -4 & 4 \\ \hline \end{array} \][/tex]
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & -2 & 0 & 2 & 4 \\ \hline \end{array} \][/tex]
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & -4 & -8 & -10 & -10 \\ \hline \end{array} \][/tex]
Let's examine each table step-by-step.
### Table 1
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & 1 & 2 & 4 & 8 \\ \hline \end{array} \][/tex]
First differences:
[tex]\[ 2 - 1 = 1, \quad 4 - 2 = 2, \quad 8 - 4 = 4 \][/tex]
Second differences:
[tex]\[ 2 - 1 = 1, \quad 4 - 2 = 2 \][/tex]
The second differences are not constant (not all equal), so this table does not represent a quadratic relationship.
### Table 2
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & -2 & -4 & -8 & -16 \\ \hline \end{array} \][/tex]
First differences:
[tex]\[ -4 - (-2) = -2, \quad -8 - (-4) = -4, \quad -16 - (-8) = -8 \][/tex]
Second differences:
[tex]\[ -4 - (-2) = -2, \quad -8 - (-4) = -4 \][/tex]
The second differences are not constant, so this table does not represent a quadratic relationship.
### Table 3
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & 3 & 4 & 5 & 6 \\ \hline \end{array} \][/tex]
First differences:
[tex]\[ 4 - 3 = 1, \quad 5 - 4 = 1, \quad 6 - 5 = 1 \][/tex]
Second differences:
[tex]\[ 1 - 1 = 0, \quad 1 - 1 = 0 \][/tex]
The second differences are constant (all zero), indicating that this table represents a quadratic relationship.
### Table 4
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & 4 & -4 & -4 & 4 \\ \hline \end{array} \][/tex]
First differences:
[tex]\[ -4 - 4 = -8, \quad -4 - (-4) = 0, \quad 4 - (-4) = 8 \][/tex]
Second differences:
[tex]\[ 0 - (-8) = 8, \quad 8 - 0 = 8 \][/tex]
The second differences are constant (all equal to 8), indicating that this table represents a quadratic relationship.
### Table 5
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & -2 & 0 & 2 & 4 \\ \hline \end{array} \][/tex]
First differences:
[tex]\[ 0 - (-2) = 2, \quad 2 - 0 = 2, \quad 4 - 2 = 2 \][/tex]
Second differences:
[tex]\[ 2 - 2 = 0, \quad 2 - 2 = 0 \][/tex]
The second differences are constant (all zero), indicating that this table represents a quadratic relationship.
### Table 6
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & -4 & -8 & -10 & -10 \\ \hline \end{array} \][/tex]
First differences:
[tex]\[ -8 - (-4) = -4, \quad -10 - (-8) = -2, \quad -10 - (-10) = 0 \][/tex]
Second differences:
[tex]\[ -2 - (-4) = 2, \quad 0 - (-2) = 2 \][/tex]
The second differences are constant (all equal to 2), indicating that this table represents a quadratic relationship.
Based on this analysis, the tables representing quadratic relationships are:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & 3 & 4 & 5 & 6 \\ \hline \end{array} \][/tex]
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & 4 & -4 & -4 & 4 \\ \hline \end{array} \][/tex]
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & -2 & 0 & 2 & 4 \\ \hline \end{array} \][/tex]
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & -4 & -8 & -10 & -10 \\ \hline \end{array} \][/tex]
