Let's analyze how the [tex]\( y \)[/tex] values change as the [tex]\( x \)[/tex] values increase.
Given the data:
[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]
We can calculate the changes in the [tex]\( y \)[/tex] values for each interval:
- From [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex], the [tex]\( y \)[/tex] value changes from 2 to 4.
- From [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex], the [tex]\( y \)[/tex] value changes from 4 to 8.
- From [tex]\( x = 3 \)[/tex] to [tex]\( x = 4 \)[/tex], the [tex]\( y \)[/tex] value changes from 8 to 16.
- From [tex]\( x = 4 \)[/tex] to [tex]\( x = 5 \)[/tex], the [tex]\( y \)[/tex] value changes from 16 to 32.
Next, let's observe the pattern of change in [tex]\( y \)[/tex]:
1. [tex]\( 4 / 2 = 2 \)[/tex]
2. [tex]\( 8 / 4 = 2 \)[/tex]
3. [tex]\( 16 / 8 = 2 \)[/tex]
4. [tex]\( 32 / 16 = 2 \)[/tex]
In each interval, the [tex]\( y \)[/tex] value is being multiplied by 2. This indicates that the [tex]\( y \)[/tex] values are consistently multiplied by 2 as the [tex]\( x \)[/tex] values increase by 1.
Thus, the best description of how the [tex]\( y \)[/tex] values are changing over each interval is:
[tex]\[
\boxed{\text{They are being multiplied by 2 each time.}}
\][/tex]
