1. **State the problem:** We are given the function $y = -2|x| + 1$ and want to understand its graph and key features. 2. **Formula and rules:** The function involves the absolute value $|x|$, which outputs the distance of $x$ from zero, always non-negative. 3. **Analyze the function:** The coefficient $-2$ before $|x|$ means the graph is vertically stretched by a factor of 2 and reflected downward. 4. **Vertex:** The vertex (apex) of the graph is at $x=0$, so $y = -2|0| + 1 = 1$. Thus, the vertex is at $(0,1)$. 5. **Shape:** The graph forms a "V" shape opening downward because of the negative coefficient. 6. **Intercepts:** - **Y-intercept:** At $x=0$, $y=1$. - **X-intercepts:** Solve $-2|x| + 1 = 0$: $$-2|x| + 1 = 0$$ $$-2|x| = -1$$ $$|x| = \frac{1}{2}$$ So, $x = \pm \frac{1}{2}$. 7. **Summary:** The graph has vertex at $(0,1)$, opens downward, crosses the x-axis at $x=\pm \frac{1}{2}$, and y-axis at $y=1$. Final answer: The function $y = -2|x| + 1$ is a downward opening V-shaped graph with vertex at $(0,1)$ and x-intercepts at $\pm \frac{1}{2}$.