To determine how the [tex]\( y \)[/tex]-values in the table grow, let's analyze the ratio in which [tex]\( y \)[/tex]-values increase as [tex]\( x \)[/tex]-values increase.
First, let's examine the pairs of consecutive [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values given:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 1 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 49 \)[/tex]
- When [tex]\( x = 4 \)[/tex], [tex]\( y = 2401 \)[/tex]
- When [tex]\( x = 6 \)[/tex], [tex]\( y = 117649 \)[/tex]
We'll compare each successive [tex]\( y \)[/tex]-value with its preceding value:
1. From [tex]\( x = 0 \)[/tex] to [tex]\( x = 2 \)[/tex]:
[tex]\[
\frac{y(2)}{y(0)} = \frac{49}{1} = 49
\][/tex]
2. From [tex]\( x = 2 \)[/tex] to [tex]\( x = 4 \)[/tex]:
[tex]\[
\frac{y(4)}{y(2)} = \frac{2401}{49} = 49
\][/tex]
3. From [tex]\( x = 4 \)[/tex] to [tex]\( x = 6 \)[/tex]:
[tex]\[
\frac{y(6)}{y(4)} = \frac{117649}{2401} = 49
\][/tex]
From these calculations, we see that each step (increase in [tex]\( x \)[/tex] by 2) results in the [tex]\( y \)[/tex]-value being multiplied by 49.
Now, because the [tex]\( x \)[/tex]-values increase by 2 each time but we want to find the factor in terms of each single unit increase in [tex]\( x \)[/tex], we need to consider this adjustment. Each multiplication factor for a [tex]\( \Delta x \)[/tex] of 2 results in:
[tex]\[
49 \approx 7^2
\][/tex]
Thus, for each unit increase in [tex]\( x \)[/tex]:
[tex]\[
y \text{ increases by a factor of } 7^1 = 7
\][/tex]
Therefore, the [tex]\( y \)[/tex]-values increase by a factor of 7 for each [tex]\( x \)[/tex] increase of 1.
Hence, the correct answer is:
- The [tex]\( y \)[/tex]-values increase by a factor of 7 for each [tex]\( x \)[/tex] increase of 1.
