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.
