To determine how the [tex]\( y \)[/tex]-values in the table grow, we need to analyze the given data points and identify the pattern.
The table is:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
0 & 1 \\
\hline
2 & 49 \\
\hline
4 & 2,401 \\
\hline
6 & 117,649 \\
\hline
\end{tabular}
\][/tex]
We can calculate the ratios between subsequent [tex]\( y \)[/tex]-values to see if there's a multiplicative factor involved:
1. The ratio between [tex]\( y \)[/tex]-values when [tex]\( x \)[/tex] increases from 0 to 2:
[tex]\[
\frac{y(2)}{y(0)} = \frac{49}{1} = 49
\][/tex]
2. The ratio between [tex]\( y \)[/tex]-values when [tex]\( x \)[/tex] increases from 2 to 4:
[tex]\[
\frac{y(4)}{y(2)} = \frac{2,401}{49} = 49
\][/tex]
3. The ratio between [tex]\( y \)[/tex]-values when [tex]\( x \)[/tex] increases from 4 to 6:
[tex]\[
\frac{y(6)}{y(4)} = \frac{117,649}{2,401} = 49
\][/tex]
From these calculations, we observe that every time [tex]\( x \)[/tex] increases by 2 units, the [tex]\( y \)[/tex]-values increase by a factor of 49.
To understand how the [tex]\( y \)[/tex]-values change for each 1 unit increase in [tex]\( x \)[/tex], we need to recognize that:
[tex]\[
49 = 7^2
\][/tex]
Since the factor of 49 applies over 2 units of [tex]\( x \)[/tex]:
[tex]\[
49 = 7^2 \implies y \text{ increases by a factor of 7 for each x increase of 1}
\][/tex]
Thus, the correct description of how the [tex]\( y \)[/tex]-values grow for each [tex]\( x \)[/tex] increase of 1 is:
The [tex]\( y \)[/tex]-values increase by a factor of 7 for each [tex]\( x \)[/tex] increase of 1.
