Completing square from standard form

Questions where a quadratic in standard form ax²+bx+c must be converted to completed square form a(x+p)²+q to find the vertex coordinates.

4 questions

Write $3 x ^ { 2 } + 6 x + 9$ in the form $$a ( x + b ) ^ { 2 } + c$$ where $a$, $b$ and $c$ are constants to be found. The point $P$ is the minimum point of $C _ { 1 }$
Deduce the coordinates of $P$. A different curve $C _ { 2 }$ has equation $$y = A x ^ { 3 } + B x ^ { 2 } + C x + D$$ where $A$, $B$, $C$ and $D$ are constants. Given that $C _ { 2 }$
- passes through $P$
- intersects the $x$-axis at $- 4 , - 2$ and 3
- find, making your method clear, the values of $A , B , C$ and $D$.
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1. Write $4 + 12 x - 3 x ^ { 2 }$ in the form $$a + b ( x + c ) ^ { 2 }$$ where $a , b$ and $c$ are constants to be found.
2. Hence, or otherwise, state the coordinates of $M$. The line $l _ { 1 }$ passes through $O$ and $M$, as shown in Figure 4.
  A line $l _ { 2 }$ touches $C$ and is parallel to $l _ { 1 }$
Find an equation for $l _ { 2 }$

Write $2 x ^ { 2 } - 12 x + 14$ in the form $$a ( x + b ) ^ { 2 } + c$$ where $a$, $b$ and $c$ are constants to be found. Given that $C$ has a minimum at the point $P$
state the coordinates of $P$ The line $l$ intersects $C$ at $( - 1,28 )$ and at $P$ as shown in Figure 5.
Find the equation of $l$ giving your answer in the form $y = m x + c$ where $m$ and $c$ are constants to be found. The finite region $R$, shown shaded in Figure 5, is bounded by the $x$-axis, $l$, the $y$-axis, and $C$.
Use inequalities to define the region $R$.

Find the transformation which maps the curve $y = x ^ { 2 }$ to the curve $y = x ^ { 2 } + 8 x - 7$.
Write down the coordinates of the turning point of $y = x ^ { 2 } + 8 x - 7$.