Question 3 - A-Level Maths

1. Express \(x ^ { 2 } + 10 x + 19\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
2. Write down the coordinates of the vertex (minimum point) of the curve with equation \(y = x ^ { 2 } + 10 x + 19\).
3. Write down the equation of the line of symmetry of the curve \(y = x ^ { 2 } + 10 x + 19\).
4. Describe geometrically the transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 10 x + 19\).
Determine the coordinates of the points of intersection of the line \(y = x + 11\) and the curve \(y = x ^ { 2 } + 10 x + 19\).