Question

Which option correctly describes the transformation from c(x)=2(x−1)−−−−−−−√+3
𝑐
(
𝑥
)
=
2
(
𝑥
−
1
)
+
3
to b(x)=10(x−1)−−−−−−−−√+3
𝑏
(
𝑥
)
=
10
(
𝑥
−
1
)
+
3
?(1 point)
Responses

horizontal compression by a factor of 15
1
5
horizontal compression by a factor of Start Fraction 1 over 5 End Fraction

horizontal compression by a factor of 5
horizontal compression by a factor of 5

horizontal stretch by a factor of 15
1
5
horizontal stretch by a factor of Start Fraction 1 over 5 End Fraction

horizontal stretch by a factor of 5

Answer 1

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 \(\frac{1}{5}\)**.