What is the order of combinations to transform f(x)=x^2 to f(x−3)+12 ?(1 point) Responses Shift to the left 3 units and shift down 12 units. Shift to the left 3 units and shift down 12 units. Shift to the left 3 units and shift up 12 units. Shift to the left 3 units and shift up 12 units. Shift to the right 3 units and shift down 12 units. Shift to the right 3 units and shift down 12 units. Shift to the right 3 units and shift up 12 units.

Question

GPT-4o mini · Answer

To transform the function $ f(x) = x^2 $ to $ f(x - 3) + 12 $, we need to analyze the transformations involved. 1. **Transforming $ f(x) $ to $ f(x - 3) $**: This part of the transformation indicates a horizontal shift. When $ x $ is replaced by $ x - 3 $, the graph shifts **to the right** by 3 units. 2. **Transforming $ f(x - 3) $ to $ f(x - 3) + 12 $**: This part adds 12 to the function, which results in a vertical shift. When you add a positive number to the function, it shifts **upward**. Thus, adding 12 shifts the graph **up** by 12 units. Putting it all together, the order of transformations is: - Shift to the right 3 units (due to $ x - 3 $) - Shift up 12 units (due to the addition of 12) So, the correct response is: **Shift to the right 3 units and shift up 12 units.**