To express the given graph as a piecewise function, we need to break it down into different parts based on where the graph changes behavior. The graph consists of two linear segments:

1. The first segment extends from $(-6, -4)$ to $(-3, -2)$.
2. The second segment extends from $(-3, 2)$ to $(4, 9)$.

Step 1: Find the equations of the lines for each segment.

Segment 1: From (-6, -4) to (-3, -2)

We use the slope-intercept form $y = mx + b$, where $m$ is the slope.

• Slope (m): $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - (-4)}{-3 - (-6)} = \frac{-2 + 4}{-3 + 6} = \frac{2}{3}$

Now, using the point-slope form with the point $(-6, -4)$:

$y + 4 = \frac{2}{3}(x + 6)$

Simplifying:

$y = \frac{2}{3}(x + 6) - 4$

Distribute:

$y = \frac{2}{3}x + 4 - 4 = \frac{2}{3}x$

Thus, the equation for this segment is $y = \frac{2}{3}x$, valid for $x \in [-6, -3]$.

Segment 2: From (-3, 2) to (4, 9)

We calculate the slope again:

• Slope (m): $m = \frac{9 - 2}{4 - (-3)} = \frac{7}{7} = 1$

Using the point-slope form with the point $(-3, 2)$:

$y - 2 = 1(x + 3)$

Simplifying:

$y = x + 5$

Thus, the equation for this segment is $y = x + 5$, valid for $x \in (-3, 4]$.

Step 2: Express the Piecewise Function

Combining the two segments, the piecewise function is: