Completing the square, form and properties

2. $$f ( x ) = 3 + 12 x - 2 x ^ { 2 }$$

1. Express \(\mathrm { f } ( x )\) in the form
   2. \(\mathrm { f } ( x ) = 3 + 12 x - 2 x ^ { 2 }\)
2. Express \(\mathrm { f } ( x )\) in the form $$\begin{aligned} & \qquad a - b ( x + c ) ^ { 2 } \\ & \text { where } a , b \text { and } c \text { are integers to be found. } \\ & \text { he curve with equation } y = \mathrm { f } ( x ) - 7 \text { crosses the } x \text {-axis at the points } P \text { and } Q \text { and crosses } \\ & \text { te } y \text {-axis at the point } R \text {. } \\ & \text { F) Find the area of the triangle } P Q R \text {, giving your answer in the form } m \sqrt { n } \text { where } m \text { and } \\ & n \text { are integers to be found. } \end{aligned}$$ \(\_\_\_\_\) "

9 The graph shows the function \(y = x ^ { 2 } + b x + c\) where \(b\) and \(c\) are constants.
The point \(\mathrm { M } ( - 3 , - 16 )\) on the graph is the minimum point of the graph. \includegraphics[max width=\textwidth, alt={}, center]{3b6291ef-bef9-49de-a20f-591e621bed65-2_478_948_1871_588}

1. Write down the function \(y = \mathrm { f } ( x )\) in completed square form.
2. Hence find the coordinates of the points where the curve cuts the axes.

8

1. Express \(2 x ^ { 2 } + 5 x\) in the form \(2 ( x + p ) ^ { 2 } + q\).
2. State the coordinates of the minimum point of the curve \(y = 2 x ^ { 2 } + 5 x\).
3. State the equation of the normal to the curve at its minimum point.
4. Solve the inequality \(2 x ^ { 2 } + 5 x > 0\).

5

1. 1. Express \(2 x ^ { 2 } + 6 x + 5\) in the form \(2 ( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
   2. Hence write down the minimum value of \(2 x ^ { 2 } + 6 x + 5\).
2. The point \(A\) has coordinates \(( - 3,5 )\) and the point \(B\) has coordinates \(( x , 3 x + 9 )\).
   1. Show that \(A B ^ { 2 } = 5 \left( 2 x ^ { 2 } + 6 x + 5 \right)\).
   2. Use your result from part (a)(ii) to find the minimum value of the length \(A B\) as \(x\) varies, giving your answer in the form \(\frac { 1 } { 2 } \sqrt { n }\), where \(n\) is an integer.

6

1. 1. Express \(x ^ { 2 } - 8 x + 17\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
   2. Hence write down the minimum value of \(x ^ { 2 } - 8 x + 17\).
   3. State the value of \(x\) for which the minimum value of \(x ^ { 2 } - 8 x + 17\) occurs.
      (1 mark)
2. The point \(A\) has coordinates (5,4) and the point \(B\) has coordinates ( \(x , 7 - x\) ).
   1. Expand \(( x - 5 ) ^ { 2 }\).
   2. Show that \(A B ^ { 2 } = 2 \left( x ^ { 2 } - 8 x + 17 \right)\).
   3. Use your results from part (a) to find the minimum value of the distance \(A B\) as \(x\) varies.