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To solve these problems, we need to perform a dilation on each shape. A dilation changes the size of a shape but keeps its proportions and orientation the same.

The rule for dilating a point $(x, y)$ about the origin $(0,0)$ with a scale factor of $k$ is:
$$ (x, y) \rightarrow (k \cdot x, k \cdot y) $$

This means you simply multiply the x-coordinate and the y-coordinate by the scale factor.

Here is the step-by-step solution for each problem:

1. Quadrilateral with Scale Factor 3

First, identify the coordinates of the vertices of the original quadrilateral from the graph:
* Top-left vertex: $(-4, 2)$
* Bottom-left vertex: $(-4, -4)$
* Bottom-right vertex: $(1, -3)$
* Top-right vertex: $(4, 3)$

Now, multiply each coordinate by the scale factor of 3:
* $(-4, 2) \rightarrow (-4 \cdot 3, 2 \cdot 3) = \mathbf{(-12, 6)}$
* $(-4, -4) \rightarrow (-4 \cdot 3, -4 \cdot 3) = \mathbf{(-12, -12)}$
* $(1, -3) \rightarrow (1 \cdot 3, -3 \cdot 3) = \mathbf{(3, -9)}$
* $(4, 3) \rightarrow (4 \cdot 3, 3 \cdot 3) = \mathbf{(12, 9)}$

Action: Plot these four new points and connect them to form the larger quadrilateral.

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2. Triangle with Scale Factor $\frac{1}{3}$

Identify the coordinates of the triangle's vertices:
* Left vertex: $(-9, 3)$
* Bottom vertex: $(-3, -6)$
* Right vertex: $(6, -6)$

Multiply each coordinate by the scale factor of $\frac{1}{3}$ (which is the same as dividing by 3):
* $(-9, 3) \rightarrow (-9 \div 3, 3 \div 3) = \mathbf{(-3, 1)}$
* $(-3, -6) \rightarrow (-3 \div 3, -6 \div 3) = \mathbf{(-1, -2)}$
* $(6, -6) \rightarrow (6 \div 3, -6 \div 3) = \mathbf{(2, -2)}$

Action: Plot these three new points and connect them to form the smaller triangle.

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3. Quadrilateral with Scale Factor $\frac{1}{2}$

Identify the coordinates of the quadrilateral's vertices:
* Top-left vertex: $(-6, 4)$
* Bottom-left vertex: $(-6, -6)$
* Middle vertex: $(2, 0)$
* Top-right vertex: $(4, 6)$

Multiply each coordinate by the scale factor of $\frac{1}{2}$ (which is the same as dividing by 2):
* $(-6, 4) \rightarrow (-6 \div 2, 4 \div 2) = \mathbf{(-3, 2)}$
* $(-6, -6) \rightarrow (-6 \div 2, -6 \div 2) = \mathbf{(-3, -3)}$
* $(2, 0) \rightarrow (2 \div 2, 0 \div 2) = \mathbf{(1, 0)}$
* $(4, 6) \rightarrow (4 \div 2, 6 \div 2) = \mathbf{(2, 3)}$

Action: Plot these four new points and connect them to form the smaller quadrilateral.

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4. Quadrilateral with Scale Factor 4

Identify the coordinates of the quadrilateral's vertices:
* Top-left vertex: $(-3, 3)$
* Bottom-left vertex: $(-2, -3)$
* Right vertex: $(1, 0)$
* Top-right vertex: $(1, 2)$

Multiply each coordinate by the scale factor of 4:
* $(-3, 3) \rightarrow (-3 \cdot 4, 3 \cdot 4) = \mathbf{(-12, 12)}$
* $(-2, -3) \rightarrow (-2 \cdot 4, -3 \cdot 4) = \mathbf{(-8, -12)}$
* $(1, 0) \rightarrow (1 \cdot 4, 0 \cdot 4) = \mathbf{(4, 0)}$
* $(1, 2) \rightarrow (1 \cdot 4, 2 \cdot 4) = \mathbf{(4, 8)}$

Action: Plot these four new points and connect them to form the larger quadrilateral. Note that the top-left point $(-12, 12)$ will be at the very edge of the grid provided in the worksheet.

Final Answer:
1. New Vertices: $(-12, 6), (-12, -12), (3, -9), (12, 9)$
2. New Vertices: $(-3, 1), (-1, -2), (2, -2)$
3. New Vertices: $(-3, 2), (-3, -3), (1, 0), (2, 3)$
4. New Vertices: $(-12, 12), (-8, -12), (4, 0), (4, 8)$