1. **State the problem:** We need to express the graph as a piecewise function based on the two line segments given. 2. **Identify the points and intervals:** - First segment: from $x = -4$ (open circle, so not included) to $x = 0$ (closed circle, included), points $(-4, 2)$ to $(0, 5)$. - Second segment: from $x = 0$ (closed circle) to $x = 6$ (closed circle), points $(0, 5)$ to $(6, 7)$. 3. **Find the equation of the first line segment:** - Slope formula: $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 2}{0 - (-4)} = \frac{3}{4}$$ - Equation form: $$y = mx + b$$ - Use point $(0,5)$ to find $b$: $$5 = \frac{3}{4} \times 0 + b \Rightarrow b = 5$$ - So, first line segment equation: $$y = \frac{3}{4}x + 5$$ 4. **Find the equation of the second line segment:** - Points: $(0,5)$ and $(6,7)$ - Slope: $$m = \frac{7 - 5}{6 - 0} = \frac{2}{6} = \frac{1}{3}$$ - Use point $(0,5)$ to find $b$: $$5 = \frac{1}{3} \times 0 + b \Rightarrow b = 5$$ - So, second line segment equation: $$y = \frac{1}{3}x + 5$$ 5. **Write the piecewise function:** $$ f(x) = \begin{cases} \frac{3}{4}x + 5 & \text{for } -4 < x \leq 0 \\ \frac{1}{3}x + 5 & \text{for } 0 < x \leq 6 \end{cases} $$ 6. **Explanation:** - The first segment excludes $x = -4$ because of the open circle, so $x > -4$. - The point $x=0$ is included in both segments as a closed circle. - The second segment includes $x=6$ as a closed circle. Final answer: $$ f(x) = \begin{cases} \frac{3}{4}x + 5 & -4 < x \leq 0 \\ \frac{1}{3}x + 5 & 0 < x \leq 6 \end{cases} $$