Quadratic function range and roots analysis

A question is this type if and only if it asks about the range of a quadratic function, conditions on constants for real/repeated roots, or properties of roots (e.g. discriminant conditions, sum and product of roots).

2 questions · Moderate -0.6

1.02e Complete the square: quadratic polynomials and turning points
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CAIE P1 2020 Specimen Q11
9 marks Moderate -0.8
11 The function f is defined, for \(x \in \mathbb { R }\), by \(\mathrm { f } : x \mapsto x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants.
  1. It is given that \(a = 6\) and \(b = - 8\). Find the range of f .
  2. It is given instead that \(a = 5\) and that the roots of the equation \(\mathrm { f } ( x ) = 0\) are \(k\) and \(- 2 k\), where \(k\) is a constant. Find the values of \(b\) and \(k\).
  3. Show that if the equation \(\mathrm { f } ( x + a ) = a\) has no real roots then \(a ^ { 2 } < 4 ( b - a )\).
OCR H240/01 2018 June Q9
7 marks Moderate -0.3
9 The function f is defined for all real values of \(x\) as \(\mathrm { f } ( x ) = c + 8 x - x ^ { 2 }\), where \(c\) is a constant.
  1. Given that the range of f is \(\mathrm { f } ( x ) \leqslant 19\), find the value of \(c\).
  2. Given instead that \(\mathrm { ff } ( 2 ) = 8\), find the possible values of \(c\).