Standard +0.3 This is a straightforward modulus question requiring a standard V-shaped sketch and solving by cases. Part (a) is routine (intercepts at ±k/2 and -k). Part (b) involves splitting into two cases (x≥0 and x<0) and solving linear equations, which is a standard technique. The presence of parameter k adds minor algebraic complexity but no conceptual difficulty beyond typical A-level.
8. Given that \(k\) is a positive constant,
a) sketch the graph with equation
$$y = 2 | x | - k$$
Show on your sketch the coordinates of each point at which the graph crosses the \(x\)-axis and the \(y\)-axis
b) find, in terms of \(k\), the values of \(x\) for which
$$2 | x | - k = \frac { 1 } { 2 } x + \frac { 1 } { 4 } k$$
8. Given that $k$ is a positive constant,\\
a) sketch the graph with equation
$$y = 2 | x | - k$$
Show on your sketch the coordinates of each point at which the graph crosses the $x$-axis and the $y$-axis\\
b) find, in terms of $k$, the values of $x$ for which
$$2 | x | - k = \frac { 1 } { 2 } x + \frac { 1 } { 4 } k$$
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\hfill \mbox{\textit{SPS SPS FM 2022 Q8 [5]}}