Piecewise Graph Ac9247

1. **State the problem:** We need to graph the piecewise function: $$f(x) = \begin{cases} 2x + 5 & \text{for } x \leq -1 \\ -2 & \text{for } x > 5 \end{cases}$$ 2. **Analyze each piece:** - For $$x \leq -1$$, the function is a line with slope $$2$$ and y-intercept $$5$$. - For $$x > 5$$, the function is a constant $$-2$$. 3. **Find the endpoint for the first piece:** At $$x = -1$$, $$f(-1) = 2(-1) + 5 = -2 + 5 = 3$$ So the first piece ends at the closed point $$(-1, 3)$$. 4. **Graph the first piece:** Draw the line $$y = 2x + 5$$ for all $$x \leq -1$$, ending at the closed point $$(-1, 3)$$. 5. **Graph the second piece:** For $$x > 5$$, $$f(x) = -2$$ is a horizontal line. At $$x = 5$$, the value is not included, so the point $$ (5, -2) $$ is open. Draw a horizontal ray starting at $$x=5$$ (open point) extending to the right at $$y = -2$$. 6. **Summary:** - Line with slope $$2$$ for $$x \leq -1$$ ending at $$(-1, 3)$$ closed. - Horizontal ray at $$y = -2$$ for $$x > 5$$ starting at $$ (5, -2) $$ open. This completes the graphing of the piecewise function.