To analyze the transformation from \( c(x) = \sqrt{2}(x - 1) + 3 \) to \( b(x) = \sqrt{10}(x - 1) + 3 \), we can focus on the coefficients in front of the \((x - 1)\) term.
The coefficient of \((x - 1)\) in \( c(x) \) is \(\sqrt{2}\) and in \( b(x) \) it is \(\sqrt{10}\).
To understand the transformation, we can compare the two coefficients. The relationship between the two coefficients can be expressed as follows:
- The larger the coefficient, the steeper the slope and the more vertically stretch the graph.
- The transformation can be analyzed in terms of horizontal compression or stretch, based on the multiplier effect caused by the square root factor.
Calculating the ratio between the coefficients:
\[ \frac{\sqrt{10}}{\sqrt{2}} = \sqrt{\frac{10}{2}} = \sqrt{5} \]
When comparing the coefficients, the transformation does not directly equate to a simple horizontal stretch or compression. Rather, the transformation of coefficients indicates that to go from \(\sqrt{2}\) to \(\sqrt{10}\), we are essentially increasing the slope. This results in a horizontal compression because the increase in the coefficient leads to a steeper graph.
Specifically, since \( \sqrt{10} = \sqrt{2} \cdot \sqrt{5} \), we can denote that there is a horizontal compression by the factor of \( \sqrt{5} \) since we are multiplying by a factor greater than 1.
Based on typical terminology, however, there is no direct option offered that fits this continuous evaluation.
But if we consider the options given, particularly:
- "horizontal compression by a factor of 5"
This is a misrepresentation in scaling but might conceptually relate to the adjustments we see, as we move from a smaller coefficient to a larger one.
Therefore, the closest correct option that fits the analysis would be:
horizontal compression by a factor of 5.