Answer :
To determine which table represents points on the graph of the function [tex]\( h(x) = \sqrt[3]{-x+2} \)[/tex], we will compare the given points with the values calculated from the function.
Let’s analyze each table:
### Table 1:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -8 & -1 & 0 & 1 & 8 \\ \hline y & 4 & 3 & 2 & 1 & 0 \\ \hline \end{array} \][/tex]
- When [tex]\( x = -8 \)[/tex], [tex]\( h(-8) = \sqrt[3]{10} \approx 2.154 \)[/tex]
- When [tex]\( x = -1 \)[/tex], [tex]\( h(-1) = \sqrt[3]{3} \approx 1.442 \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( h(0) = \sqrt[3]{2} \approx 1.260 \)[/tex]
- When [tex]\( x = 1 \)[/tex], [tex]\( h(1) = \sqrt[3]{1} = 1 \)[/tex]
- When [tex]\( x = 8 \)[/tex], [tex]\( h(8) = \sqrt[3]{-6} \approx -1.817 \)[/tex]
The calculated values do not match the [tex]\( y \)[/tex]-values in Table 1.
### Table 2:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline y & 10 & 3 & 2 & 1 & -6 \\ \hline \end{array} \][/tex]
- When [tex]\( x = -2 \)[/tex], [tex]\( h(-2) = \sqrt[3]{4} \approx 1.587 \)[/tex]
- When [tex]\( x = -1 \)[/tex], [tex]\( h(-1) = \sqrt[3]{3} \approx 1.442 \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( h(0) = \sqrt[3]{2} \approx 1.260 \)[/tex]
- When [tex]\( x = 1 \)[/tex], [tex]\( h(1) = \sqrt[3]{1} = 1 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( h(2) = \sqrt[3]{0} = 0 \)[/tex]
The calculated values do not match the [tex]\( y \)[/tex]-values in Table 2.
### Table 3:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -6 & 1 & 2 & 3 & 10 \\ \hline y & 2 & 1 & 0 & -1 & -2 \\ \hline \end{array} \][/tex]
- When [tex]\( x = -6 \)[/tex], [tex]\( h(-6) = \sqrt[3]{8} = 2 \)[/tex]
- When [tex]\( x = 1 \)[/tex], [tex]\( h(1) = \sqrt[3]{1} = 1 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( h(2) = \sqrt[3]{0} = 0 \)[/tex]
- When [tex]\( x = 3 \)[/tex], [tex]\( h(3) = \sqrt[3]{-1} \approx -1 \)[/tex]
- When [tex]\( x = 10 \)[/tex], [tex]\( h(10) = \sqrt[3]{-8} = -2 \)[/tex]
The calculated values match the [tex]\( y \)[/tex]-values in Table 3.
### Table 4:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -4 & -3 & -2 & -1 & 0 \\ \hline y & -8 & -1 & 0 & 1 & 8 \\ \hline \end{array} \][/tex]
- When [tex]\( x = -4 \)[/tex], [tex]\( h(-4) = \sqrt[3]{6} \approx 1.817 \)[/tex]
- When [tex]\( x = -3 \)[/tex], [tex]\( h(-3) = \sqrt[3]{5} \approx 1.710 \)[/tex]
- When [tex]\( x = -2 \)[/tex], [tex]\( h(-2) = \sqrt[3]{4} \approx 1.587 \)[/tex]
- When [tex]\( x = -1 \)[/tex], [tex]\( h(-1) = \sqrt[3]{3} \approx 1.442 \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( h(0) = \sqrt[3]{2} \approx 1.260 \)[/tex]
The calculated values do not match the [tex]\( y \)[/tex]-values in Table 4.
After comparing the [tex]\( y \)[/tex]-values calculated from [tex]\( h(x) \)[/tex] with those provided in the tables, we find that Table 3 is the only one that represents points on the graph of [tex]\( h(x) = \sqrt[3]{-x+2} \)[/tex].
Let’s analyze each table:
### Table 1:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -8 & -1 & 0 & 1 & 8 \\ \hline y & 4 & 3 & 2 & 1 & 0 \\ \hline \end{array} \][/tex]
- When [tex]\( x = -8 \)[/tex], [tex]\( h(-8) = \sqrt[3]{10} \approx 2.154 \)[/tex]
- When [tex]\( x = -1 \)[/tex], [tex]\( h(-1) = \sqrt[3]{3} \approx 1.442 \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( h(0) = \sqrt[3]{2} \approx 1.260 \)[/tex]
- When [tex]\( x = 1 \)[/tex], [tex]\( h(1) = \sqrt[3]{1} = 1 \)[/tex]
- When [tex]\( x = 8 \)[/tex], [tex]\( h(8) = \sqrt[3]{-6} \approx -1.817 \)[/tex]
The calculated values do not match the [tex]\( y \)[/tex]-values in Table 1.
### Table 2:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline y & 10 & 3 & 2 & 1 & -6 \\ \hline \end{array} \][/tex]
- When [tex]\( x = -2 \)[/tex], [tex]\( h(-2) = \sqrt[3]{4} \approx 1.587 \)[/tex]
- When [tex]\( x = -1 \)[/tex], [tex]\( h(-1) = \sqrt[3]{3} \approx 1.442 \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( h(0) = \sqrt[3]{2} \approx 1.260 \)[/tex]
- When [tex]\( x = 1 \)[/tex], [tex]\( h(1) = \sqrt[3]{1} = 1 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( h(2) = \sqrt[3]{0} = 0 \)[/tex]
The calculated values do not match the [tex]\( y \)[/tex]-values in Table 2.
### Table 3:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -6 & 1 & 2 & 3 & 10 \\ \hline y & 2 & 1 & 0 & -1 & -2 \\ \hline \end{array} \][/tex]
- When [tex]\( x = -6 \)[/tex], [tex]\( h(-6) = \sqrt[3]{8} = 2 \)[/tex]
- When [tex]\( x = 1 \)[/tex], [tex]\( h(1) = \sqrt[3]{1} = 1 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( h(2) = \sqrt[3]{0} = 0 \)[/tex]
- When [tex]\( x = 3 \)[/tex], [tex]\( h(3) = \sqrt[3]{-1} \approx -1 \)[/tex]
- When [tex]\( x = 10 \)[/tex], [tex]\( h(10) = \sqrt[3]{-8} = -2 \)[/tex]
The calculated values match the [tex]\( y \)[/tex]-values in Table 3.
### Table 4:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -4 & -3 & -2 & -1 & 0 \\ \hline y & -8 & -1 & 0 & 1 & 8 \\ \hline \end{array} \][/tex]
- When [tex]\( x = -4 \)[/tex], [tex]\( h(-4) = \sqrt[3]{6} \approx 1.817 \)[/tex]
- When [tex]\( x = -3 \)[/tex], [tex]\( h(-3) = \sqrt[3]{5} \approx 1.710 \)[/tex]
- When [tex]\( x = -2 \)[/tex], [tex]\( h(-2) = \sqrt[3]{4} \approx 1.587 \)[/tex]
- When [tex]\( x = -1 \)[/tex], [tex]\( h(-1) = \sqrt[3]{3} \approx 1.442 \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( h(0) = \sqrt[3]{2} \approx 1.260 \)[/tex]
The calculated values do not match the [tex]\( y \)[/tex]-values in Table 4.
After comparing the [tex]\( y \)[/tex]-values calculated from [tex]\( h(x) \)[/tex] with those provided in the tables, we find that Table 3 is the only one that represents points on the graph of [tex]\( h(x) = \sqrt[3]{-x+2} \)[/tex].
