To analyze the transformation from \( c(x) = 2\sqrt{x - 1} + 3 \) to \( b(x) = 10\sqrt{x - 1} + 3 \), we can focus on the coefficient of the square root function.
The general form of a transformation of the function \( y = \sqrt{x} \) can be represented as \( y = a\sqrt{x - h} + k \), where:
- \( a \) affects vertical stretching or compression,
- \( h \) affects horizontal translation,
- \( k \) affects vertical translation.
In the case of \( c(x) \):
- The coefficient of the square root is \( 2 \) (i.e., \( a = 2 \)).
For \( b(x) \):
- The coefficient of the square root is \( 10 \) (i.e., \( a = 10 \)).
To compare,
- The transformation that occurs involves changing \( a \) from \( 2 \) to \( 10 \).
The horizontal transformations (compression/stretch) are related to the coefficients inside the function. In the square root context, when you change the coefficient of \( \sqrt{x - h} \), it alters the "width" of the graph horizontally.
The relationship can be understood by the formula:
- A function \( f(k(x)) \) where \( k \) is a coefficient greater than 1 compresses the graph horizontally by a factor of \( \frac{1}{k} \).
In this context:
- To change from \( 2 \) to \( 10 \), the factor effecting the horizontal stretch/compression is \(\frac{10}{2} = 5\). This indicates that \( 2 \) (the original coefficient) is being "stretched" by a factor of \( 5 \) to become \( 10 \).
Thus, the horizontal compression is:
- By a factor of \( \frac{1}{5} \).
So the correct option would be: horizontal compression by a factor of \(\frac{1}{5}\).