1. **State the problem:** We are given the function $$y = -|x + 4| - 3$$ and asked to analyze its graph and key features. 2. **Recall the absolute value function:** The basic absolute value function is $$y = |x|$$, which forms a V shape with vertex at the origin (0,0). 3. **Transformations:** - Inside the absolute value, $$x + 4$$ shifts the graph 4 units to the left. - The negative sign outside the absolute value, $$- |x + 4|$$, reflects the graph across the x-axis, flipping it upside down. - The $$-3$$ shifts the graph down by 3 units. 4. **Vertex:** The vertex of the graph is at the point where the expression inside the absolute value is zero, i.e., $$x + 4 = 0 \Rightarrow x = -4$$. Substitute $$x = -4$$ into the function: $$y = -| -4 + 4 | - 3 = -|0| - 3 = -0 - 3 = -3$$ So the vertex is at $$(-4, -3)$$. 5. **Shape and arms:** Since the graph is a reflection of the absolute value function, it forms an upside-down V shape with vertex at $$(-4, -3)$$. - For $$x > -4$$, the function behaves as $$y = -(x + 4) - 3 = -x - 4 - 3 = -x - 7$$. - For $$x < -4$$, the function behaves as $$y = -(-(x + 4)) - 3 = x + 4 - 3 = x + 1$$. 6. **Intercepts:** - **y-intercept:** Set $$x = 0$$: $$y = -|0 + 4| - 3 = -4 - 3 = -7$$ So the y-intercept is at $$(0, -7)$$. - **x-intercepts:** Set $$y = 0$$: $$0 = -|x + 4| - 3 \Rightarrow |x + 4| = -3$$ Since absolute value cannot be negative, there are no x-intercepts. **Final answer:** The graph of $$y = -|x + 4| - 3$$ is an upside-down V with vertex at $$(-4, -3)$$, y-intercept at $$(0, -7)$$, and no x-intercepts.