Use the images to answer the question.

Question

Graph of f(x)=x−−√
:

The curve f of x is graphed in quadrant 1 of a coordinate plane. The curve starts at the origin and ascends through the points left parenthesis 1 comma 1 right parenthesis and left parenthesis 4 comma 2 right parenthesis.

Transformation 1:

The curve g of x is graphed in quadrant 2 of a coordinate plane. The curve starts at the origin and ascends through the points left parenthesis negative 1 comma 1 right parenthesis and left parenthesis negative 4 comma 2 right parenthesis.

Transformation 2:

The curve h of x is graphed in quadrant 4 of a coordinate plane. The curve starts at the origin and descends through the points left parenthesis 1 comma negative 1 right parenthesis and left parenthesis 4 comma negative 2 right parenthesis.

The first image shows the graph of the function f(x)=x−−√
. The next two images show transformations of f(x)
. Use the following choices to determine the rules used to transform f(x)
. Enter the option number that corresponds to your response.

Option #1: rx−axis

Option #2: ry−axis

Option #3: ry=x

Option #4: ry=−x

(1 point)
In Transformation 1, the rule to transform f(x)
to g(x)
is Option #
.

In Transformation 2, the rule to transform f(x)
to h(x)
is Option #
.

Answer 1

To determine the transformations made to the function \( f(x) = \sqrt{x} \) to obtain \( g(x) \) and \( h(x) \), we can analyze the behavior of the curves in the different quadrants.

Transformation 1: The curve \( g(x) \) is in quadrant 2, which means it is reflected over the y-axis. The points given are \( (-1, 1) \) and \( (-4, 2) \), which correspond to the points \( (1, 1) \) and \( (4, 2) \) of the original function \( f(x) \). This transformation indicates that the function has been reflected across the y-axis.

Thus, the rule for Transformation 1 is Option #2: ry-axis.

Transformation 2: The curve \( h(x) \) is in quadrant 4, which suggests that it has been reflected over the x-axis. The points given are \( (1, -1) \) and \( (4, -2) \), which correlate to the original points \( (1, 1) \) and \( (4, 2) \) but with the y-values negated. This transformation indicates that the function is reflected across the x-axis.

Thus, the rule for Transformation 2 is Option #1: rx-axis.

Putting it all together:

In Transformation 1, the rule to transform \( f(x) \) to \( g(x) \) is Option #2.

In Transformation 2, the rule to transform \( f(x) \) to \( h(x) \) is Option #1.