Question 4 - A-Level Maths

1. Express \(x ^ { 2 } + 2 x + 5\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
2. Hence show that \(x ^ { 2 } + 2 x + 5\) is always positive.
A curve has equation \(y = x ^ { 2 } + 2 x + 5\).
1. Write down the coordinates of the minimum point of the curve.
2. Sketch the curve, showing the value of the intercept on the \(y\)-axis.
Describe the geometrical transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 2 x + 5\).