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.
(1,3) , (2,6) , (3,9) , (4,12)
The function modeled by the pattern is
, since the slope is
and the y-intercept is
.

Answer 1

the pattern is wrong try again

Answer 2

use the function f(x)=2^x for the pattern

Answer 3

To analyze the given tile pattern and determine the function relating the image number \( x \) to the number of tiles \( y \):

We are given the following points:
- When \( x = 1 \), \( y = 3 \) → (1, 3)
- When \( x = 2 \), \( y = 6 \) → (2, 6)
- When \( x = 3 \), \( y = 9 \) → (3, 9)
- When \( x = 4 \), \( y = 12 \) → (4, 12)

### Step 1: Identify the Pattern

First, we notice that as \( x \) increases by 1, \( y \) increases by 3:
- From \( x = 1 \) to \( x = 2 \), \( y \) increases from 3 to 6 (an increase of 3).
- From \( x = 2 \) to \( x = 3 \), \( y \) increases from 6 to 9 (an increase of 3).
- From \( x = 3 \) to \( x = 4 \), \( y \) increases from 9 to 12 (an increase of 3).

The consistent increase indicates a linear relationship between \( x \) and \( y \).

### Step 2: Determine the Slope and Y-intercept

The slope \( m \) of the line can be calculated as follows:
\[ m = \frac{\Delta y}{\Delta x} = \frac{3}{1} = 3 \]

To find the y-intercept \( b \), we can use one of the points. Let's use the point \( (1, 3) \):
Using the equation of the line \( y = mx + b \):
\[
3 = 3(1) + b
\]
\[
3 = 3 + b \implies b = 3 - 3 = 0
\]

### Step 3: Write the Function

The relationship can be written as:
\[ y = 3x \]

### Summary

The function modeled by the pattern is:
\[ y = 3x \]

- The slope is \( 3 \).
- The y-intercept is \( 0 \).

Thus, the answer is:
- The function modeled by the pattern is \( y = 3x \).
- The slope is \( 3 \).
- The y-intercept is \( 0 \).

Answer 4

Let’s reevaluate the points you provided and analyze the pattern again.

The points are:
- (1, 3)
- (2, 6)
- (3, 9)
- (4, 12)

### Step 1: Analyze the Points

We see that:
- For \( x = 1 \), \( y = 3 \)
- For \( x = 2 \), \( y = 6 \)
- For \( x = 3 \), \( y = 9 \)
- For \( x = 4 \), \( y = 12 \)

### Step 2: Calculate the Differences

Let's find the differences in \( y \) as \( x \) increases:
- From \( x = 1 \) to \( x = 2 \):
\[ 6 - 3 = 3 \]
- From \( x = 2 \) to \( x = 3 \):
\[ 9 - 6 = 3 \]
- From \( x = 3 \) to \( x = 4 \):
\[ 12 - 9 = 3 \]

The differences in \( y \) are constant (3), suggesting a linear relationship.

### Step 3: Determine the Slope and Y-intercept

1. **Slope (m)**: The slope can be calculated using the difference in \( y \) over the difference in \( x \):
\[
m = \frac{\Delta y}{\Delta x} = \frac{3}{1} = 3
\]

2. **Y-intercept (b)**: To find the y-intercept, we can use one of the points. Let's use the point (1, 3) in the slope-intercept form \( y = mx + b \):
\[
3 = 3(1) + b
\]
Simplifying this gives:
\[
3 = 3 + b \implies b = 3 - 3 = 0
\]

### Conclusion

Based on the analysis, the function that models the pattern is:
\[ y = 3x \]

- The slope (m) is \( 3 \).
- The y-intercept (b) is \( 0 \).

If this still does not seem correct, please provide any additional details or insights about the pattern you are looking for.

Answer 5

Let's analyze the points you initially provided in the context of the function \( f(x) = 2^x \).

### Given Points
- \( (1, 3) \)
- \( (2, 6) \)
- \( (3, 9) \)
- \( (4, 12) \)

### Checking the Function \( f(x) = 2^x \)

The function \( f(x) = 2^x \) gives us the following values:
- \( f(1) = 2^1 = 2 \)
- \( f(2) = 2^2 = 4 \)
- \( f(3) = 2^3 = 8 \)
- \( f(4) = 2^4 = 16 \)

### Observing the Values
The function \( 2^x \) does *not* match the output values of \( y \) in your points. Given your pattern (1, 3), (2, 6), (3, 9), (4, 12), we need to derive a function that would actually describe this.

### Let's Reanalyze the Pattern
Looking closely at your original points:
- For each \( x \) value, the corresponding \( y \) appears to increment by \( 3 \) for each step of \( x \), which appears to suggest a linear relationship of the form \( y = mx + b \).

### Trying a Different Approach
Instead, let's consider the coordinates again:
- From \( (1, 3) \) to \( (2, 6) \), and so on, note that:

The pattern of \( y \) gives:
- \( 3, 6, 9, 12 \), which are uniformly increasing by 3.

This suggests a relationship:
1. **Slope**: Increase of 3 per unit increase in \( x \), therefore \( m = 3 \).
2. **Y-Intercept**: Plugging in point (1, 3) gives:
\[
3 = 3(1) + b \implies b = 0.
\]

### The Correct Function
Thus, we find that the actual function relating \( x \) and \( y \) from your points is:
\[ y = 3x \]

### Conclusion
The original points do not align with the exponential growth function \( f(x) = 2^x \). Instead, for the points you provided, the function \( y = 3x \) describes the relationship accurately, where:
- The slope is \( 3 \).
- The y-intercept is \( 0 \).

If you want to explore \( f(x) = 2^x \) further or provide modified points to analyze, please share!