The two images you uploaded relate to dilations in geometry. Let's work through the problems together.

Explanation of Dilations:

A dilation is a transformation that enlarges or reduces a figure by a scale factor, with respect to a fixed point known as the center of dilation, which in these cases is the origin $(0, 0)$.

The formula for dilation is: $(x', y') = (k \cdot x, k \cdot y)$ Where:

• $(x', y')$ are the coordinates after dilation,
• $k$ is the scale factor,
• $(x, y)$ are the original coordinates before dilation.

Now, let's go through the specific exercises from the worksheets.

First Worksheet: Graph the pre-image and the image of the figure

1. Dilation of 2 about the origin

Points:

• $V(-1, -2)$
• $W(0, 2)$
• $X(1, 2)$
• $Y(2, 1)$

For a dilation of 2, multiply both $x$ and $y$ coordinates of each point by 2:

• $V'(-2, -4)$
• $W'(0, 4)$
• $X'(2, 4)$
• $Y'(4, 2)$

2. Dilation of 5 about the origin

Points:

• $G(-1, 0)$
• $H(1, 1)$
• $I(1, 0)$

For a dilation of 5:

• $G'(-5, 0)$
• $H'(5, 5)$
• $I'(5, 0)$

3. Dilation of 0.5 about the origin

Points:

• $B(-1, 1)$
• $C(2, 2)$
• $D(1, -1)$

For a dilation of 0.5:

• $B'(-0.5, 0.5)$
• $C'(1, 1)$
• $D'(0.5, -0.5)$

4. Dilation of 2 about the origin

Points:

• $E(-2, -1)$
• $F(-2, 2)$
• $G(2, -1)$

For a dilation of 2:

• $E'(-4, -2)$
• $F'(-4, 4)$
• $G'(4, -2)$

Second Worksheet: Find the coordinates of the image after the transformation

5. Dilation of $\frac{1}{4}$ about the origin

Points:

• $B(0, -1)$
• $C(0, 2)$
• $D(1, 1)$

For a dilation of $\frac{1}{4}$:

• $B'(0, -0.25)$
• $C'(0, 0.5)$
• $D'(0.25, 0.25)$

6. Dilation of 5 about the origin

Points:

• $C(0, -1)$
• $D(-1, 1)$
• $E(1, 0)$

For a dilation of 5:

• $C'(0, -5)$
• $D'(-5, 5)$
• $E'(5, 0)$

7. Dilation of 1.5 about the origin

Points:

• $E(-2, 1)$
• $D(1, 3)$
• $C(2, -2)$

For a dilation of 1.5:

• $E'(-3, 1.5)$
• $D'(1.5, 4.5)$
• $C'(3, -3)$

8. Dilation of 2 about the origin

Points:

• $H(0, 1)$
• $G(1, 1)$
• $F(1, -1)$

For a dilation of 2:

• $H'(0, 2)$
• $G'(2, 2)$
• $F'(2, -2)$

Would you like further explanation on any of these problems? Here are 5 questions to explore the concept further:

1. How would the dilation change if the center was not the origin?
2. What happens when the scale factor is negative?
3. Can you explain the effect of a dilation with a scale factor between 0 and 1?
4. How do dilations affect the area of a figure?
5. What is the relationship between the original shape and its image after dilation?

Tip:

Always multiply both the $x$ and $y$ coordinates by the scale factor to correctly calculate the image points under dilation.