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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
.

5 answers

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 \( 0 \).
the pattern is wrong try again

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.

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

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!