Question

of the following transformations, describe only the dilations, using the example: \choice j is an enlargement with a scale factor of 5, centered at the origin.\
$(x,y)\to(2/3x, 2/3y)$
$(x,y)\to(x + 2,y + 2)$
$(x,y)\to(-x,-y)$
$(x,y)\to(4x,4y)$
$(x,y)\to(y,-x)$
$(x,y)\to(x - 3,y - 3)$
$(x,y)\to(0.2x, 0.2y)$
$(x,y)\to(2x, 2y)$
an enlargement with a s.f. of 4 centered at the origin.
a rotation $270^{\circ}$ about the origin.
a reduction with a s.f. of 0.2 centered at the origin.
an enlargement with a s.f. of 4 centered at the origin.
a rotation $180^{\circ}$ about the origin.
a translation of 3 units to the left and 3 units down
a reduction with a s.f. of 2/3 centered at the origin.
an enlargement with a s.f. of 2 centered at the origin.
a reduction with a s.f. of 4 centered at the origin.
a translation of 2 units right and 2 units up.
a reduction with a s.f. of 2 centered at the origin.
an enlargement with a s.f. of 0.2 centered at the origin.
an enlargement with a s.f. of 2/3 centered at the origin.

Explanation:

Step1: Identify dilation transformations

A dilation has the form $(x,y)\to(kx,ky)$ where $k$ is the scale factor, centered at the origin. We filter the given transformations:

1. $(x,y)\to(2/3x, 2/3y)$
2. $(x,y)\to(4x, 4y)$
3. $(x,y)\to(0.2x, 0.2y)$
4. $(x,y)\to(2x, 2y)$

Step2: Match each dilation to description

• For $(x,y)\to(2/3x, 2/3y)$: $0
• For $(x,y)\to(4x, 4y)$: $k=4>1$, so it is an enlargement.
• For $(x,y)\to(0.2x, 0.2y)$: $0
• For $(x,y)\to(2x, 2y)$: $k=2>1$, so it is an enlargement.

Answer:

1. $(x,y)\to(2/3x, 2/3y)$ : A reduction with a s.f. of 2/3 centered at the origin.
2. $(x,y)\to(x+2,y+2)$ : Not a dilation (translation)
3. $(x,y)\to(-x,-y)$ : Not a dilation (rotation)
4. $(x,y)\to(4x, 4y)$ : An enlargement with a s.f. of 4 centered at the origin.
5. $(x,y)\to(y,-x)$ : Not a dilation (rotation)
6. $(x,y)\to(x-3,y-3)$ : Not a dilation (translation)
7. $(x,y)\to(0.2x, 0.2y)$ : A reduction with a s.f. of 0.2 centered at the origin.
8. $(x,y)\to(2x, 2y)$ : An enlargement with a s.f. of 2 centered at the origin.