Question

transformations in the plane 2
question 1
a single transformation in the plane has the property:

• each point a with coordinates (x, y) is transformed to a point a with coordinates (-x, y).

question 2
a single transformation in the plane has the property:

• each point b with coordinates (x, y) is transformed to a point b with coordinates (-y, x).

based on the given information, which statement is true?
a the property defines a reflection.
b the property defines a rotation.
c the property defines a translation.
d the property defines a dilation.
based on the given information, which statement is true?
a the property defines a reflection.
b the property defines a rotation.
c the property defines a translation.
d the property defines a dilation.

Explanation:

Question 1

• Reflection:
• A reflection over the \(y -\)axis changes the \(x -\)coordinate's sign. If a point \(A(x,y)\) is transformed to \(A'(-x,y)\), this is a reflection over the \(y -\)axis.
• Rotation:
• A rotation of \(90^{\circ}\) about the origin transforms a point \((x,y)\) to \((-y,x)\), a rotation of \(180^{\circ}\) transforms \((x,y)\) to \((-x,-y)\), and a rotation of \(270^{\circ}\) transforms \((x,y)\) to \((y, - x)\). Since the \(y -\)coordinate remains the same in the transformation \(A(x,y)\to A'(-x,y)\), it is not a rotation.
• Translation:
• A translation has the form \((x,y)\to(x + a,y + b)\) where \(a

eq0\) or \(b
eq0\). Here, it is not of the form \((x,y)\to(x + a,y + b)\), so it is not a translation.

• Dilation:
• A dilation has the form \((x,y)\to(kx,ky)\) where \(k

eq1\). Since it is not of the form \((x,y)\to(kx,ky)\), it is not a dilation.

Question 2

• Reflection:
• A reflection over the \(x -\)axis is \((x,y)\to(x,-y)\), over the \(y -\)axis is \((x,y)\to(-x,y)\), and over the line \(y = x\) is \((x,y)\to(y,x)\). The transformation \(B(x,y)\to B'(-y,x)\) is not a reflection.
• Rotation:
• A rotation of \(90^{\circ}\) counter - clockwise about the origin has the transformation rule \((x,y)\to(-y,x)\).
• Translation:
• A translation has the form \((x,y)\to(x + a,y + b)\) where \(a

eq0\) or \(b
eq0\). Here, it is not of the form \((x,y)\to(x + a,y + b)\), so it is not a translation.

• Dilation:
• A dilation has the form \((x,y)\to(kx,ky)\) where \(k

eq1\). Since it is not of the form \((x,y)\to(kx,ky)\), it is not a dilation.

Answer:

Question 1: A. The property defines a reflection.
Question 2: B. The property defines a rotation.