Given that a and b are positive constants, (a) on separate diagrams, sketch the graph with equation (i) y = |2x - a| (ii) y = |2x - a| + b Show, on each sketch, the coordinates of each point at which the graph crosses or meets the axes - Edexcel - A-Level Maths Pure - Question 8 - 2017 - Paper 4

For the graph of the equation $y = |2x - a| + b$, we also find the key intercepts:

1. Find x-intercept: Set $y = 0$: [ |2x - a| + b = 0 \implies |2x - a| = -b. ] Since $b > 0$, no solutions exist for the x-intercept.

2. Find y-intercept: Set $x = 0$: [ y = |2(0) - a| + b = a + b. ] Thus, the y-intercept is at $(0, a + b)$.

3. Sketch the graph: This graph is also V-shaped, shifted up by $b$. The vertex will be at $(\frac{a}{2}, b)$, hence the x-axis will not be crossed.

To find the value of $c$ in terms of $a$, we consider the equation:

[|2x - a| + b = \frac{3}{2}x + 8.]

1. Evaluate at x = 0: [|2(0) - a| + b = | - a| + b = a + b = 8.] Thus, we have: [ b = 8 - a. ]

2. Evaluate at x = c: Substitute $x = c$: [|2c - a| + (8 - a) = \frac{3}{2}c + 8.]

3. Rearranging leads to: [|2c - a| = \frac{3}{2}c + a - 8.]

4. Case Analysis: Solve the cases of $2c - a$ being positive and negative, equate and simplify accordingly to find the relation involving $c$ and $a$. After solving, it will yield: [ c = \frac{2(a - 8)}{3}. ]