Question

use the image to answer the question.
image long description
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 ( y = square ), since the slope is ( square ) and the ( y )-intercept is ( square ).

Explanation:

Step1: Count tiles for each x

For \( x = 1 \), count tiles: \( 3 \). For \( x = 2 \), count: \( 6 \). For \( x = 3 \), count: \( 9 \). For \( x = 4 \), count: \( 12 \). Wait, no, wait the first image (x=0? Wait, the first is x=0? Wait the labels: 0,1,2,3,4. Wait x=0: maybe 0? Wait no, the first figure after 0 is x=1. Wait let's re - examine.

Wait the first figure (x=1) has 3 tiles, x=2 has 6, x=3 has 9, x=4 has 12? Wait no, maybe my initial count is wrong. Wait let's look at the long description. Wait the problem says "image number" x, and y is number of tiles. Let's list x and y:

x = 1: Let's count the tiles. The first non - zero image (x=1) has 3 tiles? Wait no, looking at the image:

For x = 1: The figure has 3 tiles (two in the bottom, one above? Wait no, the first figure (x=1) has 3 tiles? Wait x=2: let's count. The second figure (x=2) has 6 tiles? Wait no, maybe the pattern is linear. Let's find the slope.

Slope \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let's take x=1, y=3; x=2, y=6. Then \( m=\frac{6 - 3}{2 - 1}=3 \).

Now, the y - intercept: when x = 0, what's y? If x=1, y=3, and slope is 3, then the equation is \( y=mx + b \). Plug x=1, y=3: \( 3 = 3\times1 + b\Rightarrow b = 0 \)? Wait no, maybe x starts at 0. Wait the first image is labeled 0, then 1,2,3,4. Let's check x=0: maybe y=0? No, wait the first figure (x=0) is empty? Then x=1: 3 tiles, x=2: 6 tiles, x=3: 9 tiles, x=4: 12 tiles. So the pattern is y = 3x.

Wait let's verify:

For x=1: \( y = 3\times1=3 \) (matches the count).

For x=2: \( y = 3\times2 = 6 \) (matches).

For x=3: \( y = 3\times3=9 \) (matches).

For x=4: \( y = 3\times4 = 12 \) (matches).

So the slope is 3, y - intercept is 0. So the function is \( y = 3x \).

Step2: Confirm the equation

The general form of a linear equation is \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. We found \( m = 3 \) and \( b = 0 \), so the equation is \( y=3x \).

Answer:

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