Define a function that transforms the parent root function with a horizontal compression by a factor of 5 and a downward shift of 10 units.

Question

(1 point)
Responses

f left-parenthesis x right-parenthesis equals Root Index n Start Root 5 x End Root minus 10
Image with alt text: f left-parenthesis x right-parenthesis equals Root Index n Start Root 5 x End Root minus 10

f left-parenthesis x right-parenthesis equals 5 Root Index n Start Root x End Root minus 10
Image with alt text: f left-parenthesis x right-parenthesis equals 5 Root Index n Start Root x End Root minus 10

f left-parenthesis x right-parenthesis equals Root Index n Start Root 5 left-parenthesis x minus 2 right-parenthesis End Root
Image with alt text: f left-parenthesis x right-parenthesis equals Root Index n Start Root 5 left-parenthesis x minus 2 right-parenthesis End Root

f left-parenthesis x right-parenthesis equals 5 Root Index n Start Root x minus 10 End Root
Image with alt text: f left-parenthesis x right-parenthesis equals 5 Root Index n Start Root x minus 10 End Root

Answer 1

To define a function that performs a horizontal compression by a factor of 5 and a downward shift of 10 units, we start with the parent root function \( f(x) = \sqrt{x} \).

Horizontal Compression

A horizontal compression by a factor of 5 is achieved by modifying the input \( x \) to \( 5x \). This means the function will become: \[ f(x) = \sqrt{5x} \]

Downward Shift

To apply a downward shift of 10 units, we subtract 10 from the entire function: \[ f(x) = \sqrt{5x} - 10 \]

Final Function

Thus, the final function that incorporates both transformations (horizontal compression by a factor of 5 and a downward shift of 10 units) is: \[ f(x) = \sqrt{5x} - 10 \]

This corresponds to the first option in the provided responses: f(x) = Root Index n Start Root 5 x End Root - 10.