Question

1. write the equation of the function g(x), in terms of x, given the information below. like the parent function $f(x) = \sqrt{x}$, but reflected in the x - axis, then translated horizontally - 2 units and vertically 1 unit.

Explanation:

Step1: Reflect over x - axis

To reflect a function \( y = f(x) \) over the \( x \) - axis, we replace \( f(x) \) with \( -f(x) \). For the parent function \( f(x)=\sqrt{x} \), after reflecting over the \( x \) - axis, the function becomes \( y =-\sqrt{x} \).

Step2: Horizontal translation

A horizontal translation of \( h \) units is given by replacing \( x \) with \( x - h \) in the function. If we want to translate the function horizontally by \( - 2\) units (which is a translation of 2 units to the left), we replace \( x \) with \( x-(-2)=x + 2 \) in the function. So the function after horizontal translation is \( y=-\sqrt{x + 2} \).

Step3: Vertical translation

A vertical translation of \( k \) units is given by adding \( k \) to the function. We want to translate the function vertically by 1 unit. So we add 1 to the function. The function becomes \( g(x)=-\sqrt{x + 2}+1 \).

Answer:

\( g(x)=-\sqrt{x + 2}+1 \)