Algebraic Representations of Dilations worksheet with graphing exercises for students to practice dilations on coordinate planes.

I can't view or analyze images directly, so I can't see the specific figures or details in your uploaded image. However, based on the title and structure of the worksheet you've described — "Algebraic Representations of Dilations" — I can help you solve typical dilation problems like those shown in such worksheets.

Let me walk through how to solve these types of problems step by step, assuming the standard format of this worksheet:

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What is a Dilation?

A dilation is a transformation that changes the size of a figure while keeping its shape and orientation. It's centered at a point (usually the origin unless otherwise specified) and uses a scale factor.

- If the scale factor \( k > 1 \), the image is larger.
- If \( 0 < k < 1 \), the image is smaller.
- If \( k < 0 \), the image is reflected across the center and scaled.

The algebraic rule for a dilation centered at the origin is:
\[
(x, y) \rightarrow (kx, ky)
\]
where \( k \) is the scale factor.

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How to Solve These Problems

#### Step 1: Identify the Coordinates of the Original Figure
Look at the graph and write down the coordinates of each vertex of the figure.

For example, suppose in Problem 3, triangle \( \triangle PQR \) has vertices:
- \( P = (2, 4) \)
- \( Q = (6, 2) \)
- \( R = (4, 8) \)

And the scale factor is \( \frac{1}{2} \).

#### Step 2: Apply the Dilation Rule
Multiply each coordinate by the scale factor.

So:
- \( P' = \left(2 \times \frac{1}{2}, 4 \times \frac{1}{2}\right) = (1, 2) \)
- \( Q' = \left(6 \times \frac{1}{2}, 2 \times \frac{1}{2}\right) = (3, 1) \)
- \( R' = \left(4 \times \frac{1}{2}, 8 \times \frac{1}{2}\right) = (2, 4) \)

#### Step 3: Plot the New Points
Graph the new points \( P', Q', R' \) on the grid to form the dilated image.

#### Step 4: Write the Answer
Fill in the blanks with the new coordinates.

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Example for Problem 4

Suppose the original quadrilateral has vertices:
- \( M = (2, 2) \)
- \( N = (6, 2) \)
- \( O = (6, 6) \)
- \( P = (2, 6) \)

Scale factor = 2

Then:
- \( M' = (2 \times 2, 2 \times 2) = (4, 4) \)
- \( N' = (6 \times 2, 2 \times 2) = (12, 4) \)
- \( O' = (6 \times 2, 6 \times 2) = (12, 12) \)
- \( P' = (2 \times 2, 6 \times 2) = (4, 12) \)

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General Tips

1. Always check if the dilation is centered at the origin.
2. Use the rule: \( (x, y) \rightarrow (kx, ky) \).
3. Be careful with negative scale factors — they reflect the image.
4. Double-check your multiplication.

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If you can describe the figures or provide the coordinates from the worksheet, I can help you solve them exactly! For now, use the method above to work through each problem on your sheet.