1. **State the problem:** We need to analyze and understand the function $$f(x) = -|2x + 1|$$ and find some values to plot its graph. 2. **Recall the absolute value function:** The absolute value function $$|u|$$ outputs the distance of $$u$$ from zero, always non-negative. Here, $$u = 2x + 1$$. 3. **Understand the negative sign:** The function $$f(x) = -|2x + 1|$$ means we take the absolute value of $$2x + 1$$ and then multiply by $$-1$$, so the output is always non-positive (zero or negative). 4. **Calculate some values:** - For $$x = -1$$: $$2(-1) + 1 = -2 + 1 = -1$$, so $$f(-1) = -|-1| = -1$$. - For $$x = 0$$: $$2(0) + 1 = 1$$, so $$f(0) = -|1| = -1$$. - For $$x = -0.5$$: $$2(-0.5) + 1 = -1 + 1 = 0$$, so $$f(-0.5) = -|0| = 0$$. 5. **Plot points:** The vertex of the V-shaped graph is at $$x = -0.5$$ where $$f(x) = 0$$. The graph opens downward because of the negative sign. 6. **Summary:** The function is a downward V-shaped graph with vertex at $$(-0.5, 0)$$ and values decreasing as $$x$$ moves away from $$-0.5$$. Final answer: $$f(x) = -|2x + 1|$$ is a V-shaped graph opening downward with vertex at $$(-0.5, 0)$$.