https://file.aiquickdraw.com/mathbot/file/1726198792923eppn8d1u.jpg

Let's analyze the image.

1. **Pattern of the dots (from the figures):**
   - Figure 1: 3 dots
   - Figure 2: 6 dots
   - Figure 3: 9 dots
   - Figure 4: 12 dots

The number of dots increases by 3 with each figure. It follows a linear relationship.

2. **Completing the table:**
   Based on the pattern:

   | Figure No. (x) | Number of Dots (y) |
   |----------------|--------------------|
   | 1              | 3                  |
   | 2              | 6                  |
   | 3              | 9                  |
   | 4              | 12                 |
   | 5              | 15                 |
   | 10             | 30                 |
   | 15             | 45                 |
   | x              | \( y = 3x \)       |

3. **Relating function:**
   The number of dots \( y \) relates to the figure number \( x \) by the function: 
   \[
   y = 3x
   \]
   This means that for each figure number \( x \), you multiply it by 3 to get the number of dots.

4. **Number of dots in the 19th figure:**
   Using the function \( y = 3x \), for \( x = 19 \):
   \[
   y = 3(19) = 57 \text{ dots}
   \]

5. **Is there a figure with 260 dots?**
   To check if there is a figure with 260 dots, solve for \( x \) when \( y = 260 \):
   \[
   260 = 3x
   \]
   \[
   x = \frac{260}{3} = 86.67
   \]
   Since \( x \) must be an integer, there is no figure with exactly 260 dots.

Would you like further details or have any questions?

### Related Questions:
1. What would the number of dots be for the 25th figure?
2. If the pattern changed to adding 4 dots per figure, what would the function be?
3. How many dots would the 50th figure have?
4. What figure would contain exactly 75 dots?
5. What happens if we plot the number of dots against the figure number? What shape would the graph have?

**Tip:** When analyzing patterns, check for constant differences between terms to identify linear relationships.

Examine the pattern of dots in the figures and complete the table, relate the function of figure number to the number of dots, find the number of dots in the 19th figure, and determine if there is a figure with 260 dots.

This math problem asks students to examine a linear pattern of dots, complete a table, and derive the function relating the figure number to the number of dots. It involves understanding arithmetic sequences and solving for specific figure numbers. Suitable for Grades 7-9 students.

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