Describe how the function \( f(x)=-4 \sqrt{x}+2 \) is a transformation of a toolkit function. This is a square root function stretched vertically by a factor of 4 , reflected vertically across the \( x \)-axis, and then shifted up 2 units. This is a square root function stretched horizontally by a factor of 4 , reflected vertically across the \( x \)-axis, and then shifted up 2 units. This is a square root function stretched vertically by a factor of 4 , reflected horizontally across the \( x \)-axis, and then shifted up 2 units. This is a square root function compressed vertically by a factor of 4 , reflected vertically across the \( x \)-axis, and then shifted up 2 units.

The function \( f(x) = -4 \sqrt{x} + 2 \) originates from the basic square root function \( \sqrt{x} \). The negative sign reflects the graph over the \( x \)-axis, creating a downward-opening curve. The coefficient of -4 stretches the function vertically by a factor of 4, meaning that for each \( x \), the output \( f(x) \) is four times that of the normal square root function and inverted due to the reflection. Lastly, the "+2" moves our graph upwards by 2 units. In practical terms, if you were to graph this transformation step by step, you would start with the basic square root function, apply the vertical reflection, stretch it taller by a factor of 4, and then elevate the entire graph by 2 units. The resulting function is a downward-opening curve, making it distinct from its original shape.