The graph shows two line segments, each with different slopes and defined over different intervals for $x$. To express this as a piecewise function, we need to:

1. Find the equations of the two lines.
2. Determine the domain for each line segment.

Left Line Segment:

• This line goes through points $(-9, 9)$ and $(-4, -4)$.

• To find the slope $m$ of the line: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 9}{-4 - (-9)} = \frac{-13}{5}$ So, the slope is $m = -\frac{13}{5}$.

• Now, using the point-slope form of the equation $y - y_1 = m(x - x_1)$, and taking $(-4, -4)$: $y + 4 = -\frac{13}{5}(x + 4)$ Expanding this: $y = -\frac{13}{5}x - \frac{52}{5} - 4 = -\frac{13}{5}x - \frac{72}{5}$ The equation of the left line segment is: $y = -\frac{13}{5}x - \frac{72}{5} \quad \text{for} \quad -9 \leq x \leq -4$

Right Line Segment:

• This line goes through points $(7, -6)$ and $(9, -9)$.

• The slope $m$ of the line is: $m = \frac{-9 - (-6)}{9 - 7} = \frac{-3}{2}$ So, the slope is $m = -\frac{3}{2}$.

• Again, using the point-slope form, and taking $(7, -6)$: $y + 6 = -\frac{3}{2}(x - 7)$ Expanding this: $y = -\frac{3}{2}x + \frac{21}{2} - 6 = -\frac{3}{2}x + \frac{21}{2} - \frac{12}{2} = -\frac{3}{2}x + \frac{9}{2}$ The equation of the right line segment is: $y = -\frac{3}{2}x + \frac{9}{2} \quad \text{for} \quad 7 \leq x \leq 9$

Final Piecewise Function:

Thus, the piecewise function for the graph is: