Sketch the graph of y = 4 - |2x - 6| Solve the inequality Solve the inequality 4 - |2x - 6| > 2 - AQA - A-Level Maths Pure - Question 4 - 2020 - Paper 1

To sketch the graph of the function, we need to determine the critical points and the overall shape of the graph. This function will produce an inverted V shape due to the absolute value.

1. Find the vertex: Set the expression inside the absolute value to zero to find the vertex.

   $0 = 2x - 6$

   $2x = 6$

   $x = 3$

   We substitute $x = 3$ into the function:

   $y = 4 - |2(3) - 6| = 4 - |0| = 4$

   Hence, the vertex is at (3, 4).

2. Find x-intercepts: Set y = 0:

   $0 = 4 - |2x - 6|$

   $|2x - 6| = 4$

   This gives two cases:

   • Case 1: $2x - 6 = 4$ leads to $2x = 10$ or $x = 5$.
   • Case 2: $2x - 6 = -4$ leads to $2x = 2$ or $x = 1.$

   Thus, the x-intercepts are at (1, 0) and (5, 0).

3. Y-intercept: Set $x = 0$:

   $y = 4 - |2(0) - 6| = 4 - | - 6| = 4 - 6 = -2.$

   So the y-intercept is at (0, -2).

The graph will have an inverted V shape pointing downwards, intersecting the y-axis at (0, -2), and the x-axis at (1, 0) and (5, 0), with the vertex at (3, 4). Labels for these points should be included on the graph.