To determine the transformations that need to be applied to the function [tex]\( g(x) = x^2 \)[/tex] to obtain the function [tex]\( f(x) = 2(x-4)^2 - 3 \)[/tex], let’s identify each transformation step-by-step:
1. Starting with [tex]\( g(x) = x^2 \)[/tex]:
- This is a basic parabola opening upwards with its vertex at the origin [tex]\((0,0)\)[/tex].
2. Transformation to [tex]\( 2x^2 \)[/tex]:
- The coefficient 2 in front of [tex]\( x^2 \)[/tex] indicates a vertical stretch by a factor of 2.
- Therefore, Vertical stretch is one transformation. (This corresponds to option A.)
3. Transformation to [tex]\( (x-4)^2 \)[/tex]:
- The expression inside the parentheses [tex]\( x-4 \)[/tex] indicates a horizontal translation to the right by 4 units.
- Therefore, Horizontal translation to the right is another transformation. (This corresponds to option F.)
4. Transformation to [tex]\( 2(x-4)^2 \)[/tex]:
- Combining the horizontal translation with the prior vertical stretch, we have [tex]\( 2(x-4)^2 \)[/tex].
5. Transformation to [tex]\( 2(x-4)^2 - 3 \)[/tex]:
- Subtracting 3 from the entire function means a vertical translation down by 3 units.
- Therefore, Vertical translation down is the final transformation. (This corresponds to option H.)
Thus, the transformations needed to obtain [tex]\( f(x) = 2(x-4)^2 - 3 \)[/tex] from [tex]\( g(x) = x^2 \)[/tex] are:
- Vertical stretch (A)
- Horizontal translation to the right (F)
- Vertical translation down (H)
### Final Answer
The correct transformations are represented by the options:
- A (Vertical stretch or shrink)
- F (Horizontal translation to the right)
- H (Vertical translation down)
So, the transformations needed are:
1. Vertical stretch (option A)
2. Horizontal translation to the right (option F)
3. Vertical translation down (option H)
