Sketch single transformation from given curve

5. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\). There is a maximum at \(( 0,0 )\), a minimum at \(( 2 , - 1 )\) and \(C\) passes through \(( 3,0 )\). On separate diagrams sketch the curve with equation

1. \(y = \mathrm { f } ( x + 3 )\),
2. \(y = \mathrm { f } ( - x )\). On each diagram show clearly the coordinates of the maximum point, the minimum point and any points of intersection with the \(x\)-axis.

8. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
The curve has a maximum point \(( - 2,5 )\) and an asymptote \(y = 1\), as shown in Figure 1. On separate diagrams, sketch the curve with equation

1. \(y = \mathrm { f } ( x ) + 2\)
2. \(y = 4 \mathrm { f } ( x )\)
3. \(y = \mathrm { f } ( \mathrm { x } + 1 )\) On each diagram, show clearly the coordinates of the maximum point and the equation of the asymptote.

8. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\).
The curve \(C\) passes through the origin and through \(( 6,0 )\).
The curve \(C\) has a minimum at the point \(( 3 , - 1 )\). On separate diagrams, sketch the curve with equation

1. \(y = \mathrm { f } ( 2 x )\),
2. \(y = - \mathrm { f } ( x )\),
3. \(y = \mathrm { f } ( x + p )\), where \(p\) is a constant and \(0 < p < 3\). On each diagram show the coordinates of any points where the curve intersects the \(x\)-axis and of any minimum or maximum points.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = g ( x )\).
The curve has a single turning point, a minimum, at the point \(M ( 4 , - 1.5 )\).
The curve crosses the \(x\)-axis at two points, \(P ( 2,0 )\) and \(Q ( 7,0 )\).
The curve crosses the \(y\)-axis at a single point \(R ( 0,5 )\).

1. State the coordinates of the turning point on the curve with equation \(y = 2 \mathrm {~g} ( x )\).
2. State the largest root of the equation $$g ( x + 1 ) = 0$$
3. State the range of values of \(x\) for which \(\mathrm { g } ^ { \prime } ( x ) \leqslant 0\) Given that the equation \(\mathrm { g } ( x ) + k = 0\), where \(k\) is a constant, has no real roots,
4. state the range of possible values for \(k\).

6. \begin{figure}[h] \end{figure} Figure 1 shows part of the graph of \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The graph consists of two line segments that meet at the point \(( 1 , a ) , a < 0\). One line meets the \(x\)-axis at \(( 3,0 )\). The other line meets the \(x\)-axis at \(( - 1,0 )\) and the \(y\)-axis at \(( 0 , b ) , b < 0\). In separate diagrams, sketch the graph with equation

1. \(y = \mathrm { f } ( x + 1 )\),
2. \(y = \mathrm { f } ( | x | )\). Indicate clearly on each sketch the coordinates of any points of intersection with the axes. Given that \(\mathrm { f } ( x ) = | x - 1 | - 2\), find
3. the value of \(a\) and the value of \(b\),
4. the value of \(x\) for which \(\mathrm { f } ( x ) = 5 x\).