Completing square for transformations

A question is this type if and only if it requires expressing a quadratic in completed square form to identify or describe transformations.

2 questions

CAIE P1 2016 June Q11
11 The function f is defined by \(\mathrm { f } : x \mapsto 6 x - x ^ { 2 } - 5\) for \(x \in \mathbb { R }\).
  1. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) \leqslant 3\).
  2. Given that the line \(y = m x + c\) is a tangent to the curve \(y = \mathrm { f } ( x )\), show that \(4 c = m ^ { 2 } - 12 m + 16\). The function g is defined by \(\mathrm { g } : x \mapsto 6 x - x ^ { 2 } - 5\) for \(x \geqslant k\), where \(k\) is a constant.
  3. Express \(6 x - x ^ { 2 } - 5\) in the form \(a - ( x - b ) ^ { 2 }\), where \(a\) and \(b\) are constants.
  4. State the smallest value of \(k\) for which g has an inverse.
  5. For this value of \(k\), find an expression for \(\mathrm { g } ^ { - 1 } ( x )\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
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Edexcel C3 Q5
5. $$\mathrm { f } ( x ) \equiv 2 x ^ { 2 } + 4 x + 2 , \quad x \in \mathbb { R } , \quad x \geq - 1 .$$
  1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\).
  2. Describe fully two transformations that would map the graph of \(y = x ^ { 2 } , x \geq 0\) onto the graph of \(y = \mathrm { f } ( x )\).
  3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
  4. Sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) on the same diagram and state the relationship between them.