Question

check your understanding 2
a reflection over the x -axis, followed by a 90° rotation counterclockwise about the origin.
reverse

Explanation:

Step1: Identify Coordinates

First, find coordinates of points \( D, E, F \) in the left graph. Let's assume grid has 1 unit per square. \( D(2,0) \), \( E(2,-3) \), \( F(-1,-3) \).

Step2: Reflect over x - axis

Reflection over \( x \) - axis: \((x,y)\to(x,-y)\). So \( D(2,0)\to(2,0) \), \( E(2,-3)\to(2,3) \), \( F(-1,-3)\to(-1,3) \).

Step3: Rotate 90° counterclockwise

Rotation 90° counterclockwise about origin: \((x,y)\to(-y,x)\). Apply to reflected points:
- \( D(2,0)\to(0,2) \)
- \( E(2,3)\to(-3,2) \)
- \( F(-1,3)\to(-3,-1) \)

Wait, but the "Reverse" graph (right) has \( D(2,0) \), \( E(2,-1) \)? Wait, maybe initial coordinates: Let's re - check. Left graph: \( D \) is at (2,0), \( E \) at (2, - 3)? No, maybe \( E \) is at (2, - 1)? Wait, maybe the left graph: \( F \) is at (-1, - 3), \( E \) at (2, - 3), \( D \) at (2,0). Then reflection over x - axis: \( F(-1,3) \), \( E(2,3) \), \( D(2,0) \). Then 90° counterclockwise rotation: \((x,y)\to(-y,x)\). So \( D(2,0)\to(0,2) \), \( E(2,3)\to(-3,2) \), \( F(-1,3)\to(-3,-1) \). But the right graph (Reverse) has \( E \) at (2, - 1)? Wait, maybe the problem is to reverse the transformation (undo the reflection and rotation). Let's reverse the operations: To reverse a 90° counterclockwise rotation, we do a 90° clockwise rotation (or 270° counterclockwise), and to reverse a reflection over x - axis, we reflect over x - axis again.

Let's take the right graph (Reverse) points: Let's assume \( D(2,0) \), \( E(2,-1) \), \( F(-1,-1) \) (from the grid). Wait, maybe the original figure (left) and after transformation (right)? Wait, the problem says "A reflection over the x - axis, followed by a 90° rotation counterclockwise about the origin" and "Reverse". So to reverse, we first reverse the 90° counterclockwise rotation (do 90° clockwise rotation, which is \((x,y)\to(y,-x)\)) and then reverse the reflection (reflect over x - axis again, \((x,y)\to(x,-y)\)).

Alternatively, maybe the question is to perform the transformation (reflection then rotation) on the left figure and see if it matches the right? Wait, the left figure: \( D(2,0) \), \( E(2,-3) \), \( F(-1,-3) \). Reflection over x - axis: \( D(2,0) \), \( E(2,3) \), \( F(-1,3) \). 90° counterclockwise rotation: \((x,y)\to(-y,x)\). So \( D(2,0)\to(0,2) \), \( E(2,3)\to(-3,2) \), \( F(-1,3)\to(-3,-1) \). But the right figure has \( D(2,0) \), \( E(2,-1) \), \( F(-1,-1) \). Wait, maybe the initial y - coordinates are different. Let's assume the left figure has \( E(2,-1) \), \( F(-1,-1) \), \( D(2,0) \). Then reflection over x - axis: \( E(2,1) \), \( F(-1,1) \), \( D(2,0) \). 90° counterclockwise rotation: \( D(2,0)\to(0,2) \), \( E(2,1)\to(-1,2) \), \( F(-1,1)\to(-1,-1) \). No, this is confusing. Maybe the key is to apply the transformation:

Reflection over x - axis: \((x,y)\to(x,-y)\). Let's take a point, say \( F \) in left graph: suppose \( F(-1,-3) \), after reflection: \( (-1,3) \). Then 90° counterclockwise rotation: \((x,y)\to(-y,x)\), so \( (-1,3)\to(-3,-1) \). \( E(2,-3)\to(2,3)\to(-3,2) \). \( D(2,0)\to(2,0)\to(0,2) \). Now, looking at the right graph, if we compare, maybe the right graph is the result? Wait, the problem might be to describe the transformation or find the image. But since the user hasn't fully stated the question (like "Find the image of the triangle after the transformation" or "Verify the transformation"), but assuming we need to apply the transformation:

Step1: Reflect over x - axis

For a point \((x,y)\), reflection over \( x \) - axis is \((x,-y)\). Let's take the left triangle with vertices \( D(2,0) \), \( E(2,-3) \), \( F(-1,-3) \). After reflection: \( D(2,0) \) (since \( y = 0 \)), \( E(2,3) \), \( F(-1,3) \).

Step2: Rotate 90° counterclockwise

The rule for 90° counterclockwise rotation about the origin is \((x,y)\to(-y,x)\). Applying to the reflected points:
- \( D(2,0)\): \( (-0,2)=(0,2) \)
- \( E(2,3)\): \( (-3,2) \)
- \( F(-1,3)\): \( (-3,-1) \)

So the image after reflection and rotation has vertices at \( (0,2) \), \( (-3,2) \), \( (-3,-1) \). If the right graph is supposed to be this, but maybe the grid is different. Alternatively, if the question is to reverse (undo) the transformation, we reverse the rotation (90° clockwise, \((x,y)\to(y,-x)\)) and then reverse the reflection (reflect over x - axis again).

But since the problem is about geometric transformations (reflection and rotation), the subfield is Geometry (Mathematics).

Answer:

The subfield is Geometry (Mathematics). The transformation involves reflection over \( x \) - axis (\((x,y)\to(x,-y)\)) and 90° counterclockwise rotation (\((x,y)\to(-y,x)\)) of a geometric figure (triangle with vertices \( D, E, F \)). The image after transformation can be found by applying these rules to each vertex.