Questions: The image contains a matrix or grid-like structure with numerical entries. The matrix has 2 rows and 4 columns. The entries in the top row are 1, blank, 2, blank. The entries in the bottom row are 2, 2, 2, 1. Additionally, there is a curve or path drawn over the matrix using blue lines, starting from the top-left corner and connecting (1, 2), (2, 2), (2, 2), ending at the bottom-right corner.

The image contains a matrix or grid-like structure with numerical entries. The matrix has 2 rows and 4 columns. The entries in the top row are 1, blank, 2, blank. The entries in the bottom row are 2, 2, 2, 1. Additionally, there is a curve or path drawn over the matrix using blue lines, starting from the top-left corner and connecting (1, 2), (2, 2), (2, 2), ending at the bottom-right corner.

Transcript text: The image contains a matrix or grid-like structure with numerical entries. The matrix has 2 rows and 4 columns. The entries in the top row are 1, blank, 2, blank. The entries in the bottom row are 2, 2, 2, 1. Additionally, there is a curve or path drawn over the matrix using blue lines, starting from the top-left corner and connecting (1, 2), (2, 2), (2, 2), ending at the bottom-right corner.

Solution

Solution Steps

Step 1: Identify the Coordinates of the Points

The graph shows a piecewise linear function with points at:

(-3, 1)
(-1, 2)
(1, 1)
(3, 2)

Step 2: Determine the Slopes of Each Segment

Calculate the slopes of the line segments between each pair of points:

Slope between (-3, 1) and (-1, 2): \[ \text{Slope} = \frac{2 - 1}{-1 - (-3)} = \frac{1}{2} \]
Slope between (-1, 2) and (1, 1): \[ \text{Slope} = \frac{1 - 2}{1 - (-1)} = \frac{-1}{2} \]
Slope between (1, 1) and (3, 2): \[ \text{Slope} = \frac{2 - 1}{3 - 1} = \frac{1}{2} \]

Step 3: Write the Equations of Each Line Segment

Using the point-slope form of the equation \( y - y_1 = m(x - x_1) \):

For the segment from (-3, 1) to (-1, 2): \[ y - 1 = \frac{1}{2}(x + 3) \] Simplifying: \[ y = \frac{1}{2}x + \frac{3}{2} + 1 \] \[ y = \frac{1}{2}x + \frac{5}{2} \]
For the segment from (-1, 2) to (1, 1): \[ y - 2 = \frac{-1}{2}(x + 1) \] Simplifying: \[ y = \frac{-1}{2}x - \frac{1}{2} + 2 \] \[ y = \frac{-1}{2}x + \frac{3}{2} \]
For the segment from (1, 1) to (3, 2): \[ y - 1 = \frac{1}{2}(x - 1) \] Simplifying: \[ y = \frac{1}{2}x - \frac{1}{2} + 1 \] \[ y = \frac{1}{2}x + \frac{1}{2} \]