Standard +0.3 This is a multi-part C1 question requiring standard differentiation to find a tangent, completing the square, and describing transformations. Part (iv) adds mild problem-solving by connecting previous parts, but all techniques are routine and well-practiced at this level. Slightly above average due to the synthesis required in part (iv).
9. (i) Find an equation for the tangent to the curve \(y = x ^ { 2 } + 2\) at the point \(( 1,3 )\) in the form \(y = m x + c\).
(ii) Express \(x ^ { 2 } - 6 x + 11\) in the form \(( x + a ) ^ { 2 } + b\) where \(a\) and \(b\) are integers.
(iii) Describe fully the transformation that maps the graph of \(y = x ^ { 2 } + 2\) onto the graph of \(y = x ^ { 2 } - 6 x + 11\).
(iv) Use your answers to parts (i) and (iii) to deduce an equation for the tangent to the curve \(y = x ^ { 2 } - 6 x + 11\) at the point with \(x\)-coordinate 4.
9. (i) Find an equation for the tangent to the curve $y = x ^ { 2 } + 2$ at the point $( 1,3 )$ in the form $y = m x + c$.\\
(ii) Express $x ^ { 2 } - 6 x + 11$ in the form $( x + a ) ^ { 2 } + b$ where $a$ and $b$ are integers.\\
(iii) Describe fully the transformation that maps the graph of $y = x ^ { 2 } + 2$ onto the graph of $y = x ^ { 2 } - 6 x + 11$.\\
(iv) Use your answers to parts (i) and (iii) to deduce an equation for the tangent to the curve $y = x ^ { 2 } - 6 x + 11$ at the point with $x$-coordinate 4.\\
\hfill \mbox{\textit{OCR C1 Q9 [10]}}