Answered

IDNLearn.com makes it easy to find accurate answers to your specific questions. Our community is here to provide the comprehensive and accurate answers you need to make informed decisions.

Determine the transformation(s) that need to be applied to [tex]g(x)=x^2[/tex] to obtain the graph of [tex]f(x)=2(x-4)^2-3[/tex].

Select all that apply.

A. Vertical stretch or shrink
B. Horizontal stretch or shrink
C. Reflect across the y-axis
D. Reflect across the x-axis
E. Vertical translation up
F. Horizontal translation to the right
G. Horizontal translation to the left
H. Vertical translation down

Sagot :

GinnyAnsweridn
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)
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Your questions deserve reliable answers. Thanks for visiting IDNLearn.com, and see you again soon for more helpful information.