Question

Analyze the tile pattern and write a function for the pattern. Use x for the image number and y for the number of tiles in each image. (2 points) The function modeled by the pattern is , since the slope is and the y-intercept is .

Answer 1

To analyze the tile pattern as you've described, we need to identify the relationship between the image number \( x \) and the number of tiles \( y \). From your question, it's clear that we are looking to develop a linear function in the form \( y = mx + b \), where:

- \( m \) is the slope of the line.
- \( b \) is the y-intercept (the value of \( y \) when \( x = 0 \)).

### Steps to Determine the Function:

1. **Identify Image Number (\( x \)):** This usually starts from 1 and increases by 1 for each subsequent image.

2. **Identify Number of Tiles (\( y \)):** This is the count of tiles in each corresponding image.

3. **Determine the Slope (\( m \)):** The slope can be determined by calculating the change in \( y \) (number of tiles) for a change in \( x \) (image number). If you have two points, say \((x_1, y_1)\) and \((x_2, y_2)\), the slope \( m \) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

4. **Determine the Y-Intercept (\( b \)):** The y-intercept occurs when \( x = 0 \) (though you may not have this point in your data). If you have data starting from 1, you could extend the pattern and see what value \( y \) would approach as \( x \) goes to 0, or use known values to establish a pattern.

### Example Analysis

Let’s say the pattern observed is as follows (this is a hypothetical example for illustration):

- Image 1: 3 tiles
- Image 2: 5 tiles
- Image 3: 7 tiles

Here, the following data can be compiled:

| Image Number (\( x \)) | Number of Tiles (\( y \)) |
|-----------------------|-------------------------|
| 1 | 3 |
| 2 | 5 |
| 3 | 7 |

Here, the slope \( m \) can be calculated:

- From image 1 to 2: \( m = \frac{5 - 3}{2 - 1} = \frac{2}{1} = 2 \)

And the number of tiles increases by 2 as \( x \) increases by 1. The y-intercept \( b \) in this case can be inferred if we need to extend it down to where \( x = 0 \):

By looking at the values:
1. \( y = 3 \) for \( x = 1 \)
2. If we omit one image, the intercept can be inferred by extending the pattern back to \( x = 0 \).

### Conclusion

From the example, the function modeling the pattern can be expressed as:

\[
y = 2x + 1
\]

Where:
- \( m = 2 \) (the slope)
- \( b = 1 \) (the y-intercept when extended back to \( x = 0 \))

Please provide me with particular values if you wish for real analysis on a specific problem!