Answer :
To determine how the \( y \) values are changing over each interval given in the table, let's analyze the pattern:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 2 \\ \hline 2 & 4 \\ \hline 3 & 8 \\ \hline 4 & 16 \\ \hline 5 & 32 \\ \hline \end{array} \][/tex]
1. From \( x = 1 \) to \( x = 2 \):
[tex]\[ \frac{y_{\text{at } x=2}}{y_{\text{at } x=1}} = \frac{4}{2} = 2 \][/tex]
The \( y \) value is multiplied by 2.
2. From \( x = 2 \) to \( x = 3 \):
[tex]\[ \frac{y_{\text{at } x=3}}{y_{\text{at } x=2}} = \frac{8}{4} = 2 \][/tex]
The \( y \) value is multiplied by 2.
3. From \( x = 3 \) to \( x = 4 \):
[tex]\[ \frac{y_{\text{at } x=4}}{y_{\text{at } x=3}} = \frac{16}{8} = 2 \][/tex]
The \( y \) value is multiplied by 2.
4. From \( x = 4 \) to \( x = 5 \):
[tex]\[ \frac{y_{\text{at } x=5}}{y_{\text{at } x=4}} = \frac{32}{16} = 2 \][/tex]
The \( y \) value is multiplied by 2.
In conclusion, the \( y \) values in the table are changing by being multiplied by 2 each time. Therefore, the correct description of how the \( y \) values are changing over each interval is:
They are being multiplied by 2 each time.
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 2 \\ \hline 2 & 4 \\ \hline 3 & 8 \\ \hline 4 & 16 \\ \hline 5 & 32 \\ \hline \end{array} \][/tex]
1. From \( x = 1 \) to \( x = 2 \):
[tex]\[ \frac{y_{\text{at } x=2}}{y_{\text{at } x=1}} = \frac{4}{2} = 2 \][/tex]
The \( y \) value is multiplied by 2.
2. From \( x = 2 \) to \( x = 3 \):
[tex]\[ \frac{y_{\text{at } x=3}}{y_{\text{at } x=2}} = \frac{8}{4} = 2 \][/tex]
The \( y \) value is multiplied by 2.
3. From \( x = 3 \) to \( x = 4 \):
[tex]\[ \frac{y_{\text{at } x=4}}{y_{\text{at } x=3}} = \frac{16}{8} = 2 \][/tex]
The \( y \) value is multiplied by 2.
4. From \( x = 4 \) to \( x = 5 \):
[tex]\[ \frac{y_{\text{at } x=5}}{y_{\text{at } x=4}} = \frac{32}{16} = 2 \][/tex]
The \( y \) value is multiplied by 2.
In conclusion, the \( y \) values in the table are changing by being multiplied by 2 each time. Therefore, the correct description of how the \( y \) values are changing over each interval is:
They are being multiplied by 2 each time.
