Sketch with inequalities or regions

6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 2 x + 3\).

1. Given that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\), express \(\mathrm { f } ( x )\) in a fully factorised form.
2. Sketch the graph of \(y = \mathrm { f } ( x )\), indicating the coordinates of any points of intersection with the axes.
3. Solve the inequality \(\mathrm { f } ( x ) < 0\), giving your answer in set notation.
4. The graph of \(y = \mathrm { f } ( x )\) is transformed by a stretch parallel to the \(x\)-axis, scale factor \(\frac { 1 } { 2 }\). Find the equation of the transformed graph.

1. (a) Factorise completely \(9 x - x ^ { 3 }\)

The curve \(C\) has equation $$y = 9 x - x ^ { 3 }$$ (b) Sketch \(C\) showing the coordinates of the points at which the curve cuts the \(x\)-axis. The line \(l\) has equation \(y = k\) where \(k\) is a constant.
Given that \(C\) and \(l\) intersect at 3 distinct points,
(c) find the range of values for \(k\), writing your answer in set notation. Solutions relying on calculator technology are not acceptable.

5

1. Sketch the curve \(y = \mathrm { g } ( x )\) where $$g ( x ) = ( x + 2 ) ( x - 1 ) ^ { 2 }$$ 5
2. Hence, solve \(\mathrm { g } ( x ) \leq 0\)

5

1. Sketch the curve $$y = ( x - a ) ^ { 2 } ( 3 - x ) \quad \text { where } 0 < a < 3$$ indicating the coordinates of the points where the curve and the axes meet.
   \includegraphics[max width=\textwidth, alt={}, center]{1f887565-4587-4520-99d4-f3635b015525-06_1068_1061_543_488} 5
2. Hence, solve $$( x - a ) ^ { 2 } ( 3 - x ) > 0$$ giving your answer in set notation form.