Identify the graph of $y = 1 - |x + 2|$ from the options below - AQA - A-Level Maths Mechanics - Question 1 - 2019 - Paper 2

To determine the correct graph, we first analyze the equation:

1. Understanding the Absolute Value Function: The function involves an absolute value, which indicates that there will be a V-shape in the graph.

2. Shifting the Graph: The expression $|x + 2|$ means the vertex of the V-shape will be shifted 2 units to the left on the x-axis, giving us a vertex at (-2, 1).

3. Reflection and Vertical Shift: The negative sign before the absolute value indicates a reflection over the x-axis. The graph is thus flipped upside down from its usual V-shape. The additional +1 translates the whole graph upwards by 1 unit.

4. Graph Behavior: The graph will have two line segments:

   • One line segment will extend from (-2, 1) vertically downwards.
   • The other line segment will extend upwards to both sides as x moves away from -2.

5. Comparison with Given Options: Looking at the provided graph options, only option D matches this expected behavior as it displays an inverted V-shape centered at (-2, 1).

Thus, the answer is to tick box D.