Deduce inequality solutions from sketch

8. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( 3 x - 2 ) ^ { 2 } ( x - 4 )$$

1. Deduce the values of \(x\) for which \(\mathrm { f } ( x ) > 0\)
2. Expand f(x) to the form $$a x ^ { 3 } + b x ^ { 2 } + c x + d$$ where \(a\), \(b\), \(c\) and \(d\) are integers to be found. The line \(l\), also shown in Figure 4, passes through the \(y\) intercept of \(C\) and is parallel to the \(x\)-axis. The line \(l\) cuts \(C\) again at points \(P\) and \(Q\), also shown in Figure 4 .
3. Using algebra and showing your working, find the length of line \(P Q\). Write your answer in the form \(k \sqrt { 3 }\), where \(k\) is a constant to be found.
   (Solutions relying entirely on calculator technology are not acceptable.)

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( 3 x + 20 ) ( x + 6 ) ( 2 x - 3 )$$

1. Use the given information to state the values of \(x\) for which $$f ( x ) > 0$$
2. Expand \(( 3 x + 20 ) ( x + 6 ) ( 2 x - 3 )\), writing your answer as a polynomial in simplest form. The straight line \(l\) is the tangent to \(C\) at the point where \(C\) cuts the \(y\)-axis.
   Given that \(l\) cuts \(C\) at the point \(P\), as shown in Figure 4,
3. find, using algebra, the \(x\) coordinate of \(P\)
   (Solutions based on calculator technology are not acceptable.)

10. \begin{figure}[h] \end{figure} Figure 6 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( 2 x + 5 ) ( x - 3 ) ^ { 2 }$$

1. Deduce the values of \(x\) for which \(\mathrm { f } ( x ) \leqslant 0\) The curve crosses the \(y\)-axis at the point \(P\), as shown.
2. Expand \(\mathrm { f } ( x )\) to the form $$a x ^ { 3 } + b x ^ { 2 } + c x + d$$ where \(a\), \(b\), \(c\) and \(d\) are integers to be found.
3. Hence, or otherwise, find
   1. the coordinates of \(P\),
   2. the gradient of the curve at \(P\). The curve with equation \(y = \mathrm { f } ( x )\) is translated two units in the positive \(x\) direction to a curve with equation \(y = \mathrm { g } ( x )\).
4. 1. Find \(\mathrm { g } ( x )\), giving your answer in a simplified factorised form.
   2. Hence state the \(y\) intercept of the curve with equation \(y = \mathrm { g } ( x )\).

14. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C _ { 1 }\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = ( x - 2 ) ^ { 2 } ( 2 x + 1 ) , \quad x \in \mathbb { R }$$ The curve crosses the \(x\)-axis at \(\left( - \frac { 1 } { 2 } , 0 \right)\), touches it at \(( 2,0 )\) and crosses the \(y\)-axis at ( 0,4 ). There is a maximum turning point at the point marked \(P\).

1. Use \(\mathrm { f } ^ { \prime } ( x )\) to find the exact coordinates of the turning point \(P\). A second curve \(C _ { 2 }\) has equation \(y = \mathrm { f } ( x + 1 )\).
2. Write down an equation of the curve \(C _ { 2 }\) You may leave your equation in a factorised form.
3. Use your answer to part (b) to find the coordinates of the point where the curve \(C _ { 2 }\) meets the \(y\)-axis.
4. Write down the coordinates of the two turning points on the curve \(C _ { 2 }\)
5. Sketch the curve \(C _ { 2 }\), with equation \(y = \mathrm { f } ( x + 1 )\), giving the coordinates of the points where the curve crosses or touches the \(x\)-axis.

8. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = ( x + 3 ) ^ { 2 } ( x - 1 ) , \quad x \in \mathbb { R }$$ The curve crosses the \(x\)-axis at \(( 1,0 )\), touches it at \(( - 3,0 )\) and crosses the \(y\)-axis at \(( 0 , - 9 )\)

1. In the space below, sketch the curve \(C\) with equation \(y = \mathrm { f } ( x + 2 )\) and state the coordinates of the points where the curve \(C\) meets the \(x\)-axis.
2. Write down an equation of the curve \(C\).
3. Use your answer to part (b) to find the coordinates of the point where the curve \(C\) meets the \(y\)-axis.