for the function \( g(x) \) described as follows. Use the function \( f(x)=\sqrt{x} \). Stretch the function vertically by a factor of 6 , reflect it with respect to the \( y \)-axis, and move it 9 units up.

To create the function \( g(x) \) based on \( f(x) = \sqrt{x} \), we first stretch it vertically by a factor of 6, resulting in \( 6f(x) = 6\sqrt{x} \). Next, reflecting it across the \( y \)-axis gives us \( g(-x) = 6\sqrt{-x} \). Finally, moving it 9 units up modifies the function to \( g(x) = 6\sqrt{-x} + 9 \). So, the final transformation yields: \[ g(x) = 6\sqrt{-x} + 9. \] This function is defined for \( x \leq 0 \), capturing the essence of these transformations! Transformations in functions like this one can often appear intimidating, but they can actually be visualized pretty easily! Try sketching \( f(x) \) first, then apply each transformation step-by-step on the graph, and you'll see how each alteration affects the shape and position of the curve. It’s like making your own signature style out of a basic outfit! To dive deeper into function transformations, consider resources such as "College Algebra" textbooks or online platforms like Khan Academy. They offer interactive exercises that can help you master the concept of transformations in a fun and engaging way. Remember, practice makes perfect, especially when it comes to visualizing these changes!