To find the coordinates of the pre-image of a given point using the rule [tex]\( r_y = -x(x, y) \rightarrow (-x, y) \)[/tex], let's break down this transformation step-by-step.
The given rule [tex]\( r_y = -x(x, y) \)[/tex] transforms a point [tex]\((x, y)\)[/tex] into [tex]\((-x, y)\)[/tex]. Let's determine which of the given choices will map to the point [tex]\((-4, 9)\)[/tex] under this transformation.
Here are the choices we need to evaluate:
1. [tex]\((-9, 4)\)[/tex]
2. [tex]\((-4, -9)\)[/tex]
3. [tex]\((4, 9)\)[/tex]
4. [tex]\((9, -4)\)[/tex]
We'll transform each of these points using the rule and check if it matches the image [tex]\((-4, 9)\)[/tex].
1. For the point [tex]\((-9, 4)\)[/tex]:
- Apply the transformation: [tex]\((-x, y)\)[/tex]
- Transform [tex]\((-9, 4)\)[/tex] to [tex]\((9, 4)\)[/tex]
- Since [tex]\((9, 4) \neq (-4, 9)\)[/tex], this point is not the pre-image.
2. For the point [tex]\((-4, -9)\)[/tex]:
- Apply the transformation: [tex]\((-x, y)\)[/tex]
- Transform [tex]\((-4, -9)\)[/tex] to [tex]\((4, -9)\)[/tex]
- Since [tex]\((4, -9) \neq (-4, 9)\)[/tex], this point is not the pre-image.
3. For the point [tex]\((4, 9)\)[/tex]:
- Apply the transformation: [tex]\((-x, y)\)[/tex]
- Transform [tex]\((4, 9)\)[/tex] to [tex]\((-4, 9)\)[/tex]
- Since [tex]\((-4, 9) = (-4, 9)\)[/tex], this point is the pre-image we are looking for.
4. For the point [tex]\((9, -4)\)[/tex]:
- Apply the transformation: [tex]\((-x, y)\)[/tex]
- Transform [tex]\((9, -4)\)[/tex] to [tex]\((-9, -4)\)[/tex]
- Since [tex]\((-9, -4) \neq (-4, 9)\)[/tex], this point is not the pre-image.
Therefore, the coordinates of the pre-image that transform into the point [tex]\((-4, 9)\)[/tex] are [tex]\((4, 9)\)[/tex].
