Forward transformation (given original point)

2. The point \(P ( 2,3 )\) lies on the curve with equation \(y = \mathrm { f } ( x )\). State the coordinates of the image of \(P\) under the transformation represented by the curve with equation

1. \(y = \mathrm { f } ( x + 2 )\)
2. \(y = - \mathrm { f } ( x )\)
3. \(2 y = f ( x )\)
4. \(y = \mathrm { f } ( x ) - 4\)
   State the coordinates of the image of \(P\) under the transformation represented by the curve
   with equation (a) \(y = \mathrm { f } ( x + 2 )\)

1. The point \(P ( - 4 , - 3 )\) lies on the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\)

Find the point to which \(P\) is mapped when the curve with equation \(y = \mathrm { f } ( x )\) is transformed to the curve with equation

1. \(y = \mathrm { f } ( 2 x )\)
2. \(y = 3 \mathrm { f } ( x - 1 )\)
3. \(y = | f ( x ) |\)

5
\includegraphics[max width=\textwidth, alt={}, center]{82ae6eec-3007-467c-90df-18f2adb9ccb1-2_634_926_1242_612} The graph of \(y = \mathrm { f } ( x )\) for \(- 1 \leqslant x \leqslant 4\) is shown above.

1. Sketch the graph of \(y = - \mathrm { f } ( x )\) for \(- 1 \leqslant x \leqslant 4\).
2. The point \(P ( 1,1 )\) on \(y = \mathrm { f } ( x )\) is transformed to the point \(Q\) on \(y = 3 \mathrm { f } ( x )\). State the coordinates of \(Q\).
3. Describe the transformation which transforms the graph of \(y = \mathrm { f } ( x )\) to the graph of \(y = \mathrm { f } ( x + 2 )\).

10 The diagram shows the graph of \(y = \mathrm { f } ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{4c556b8e-1a19-4480-bf2a-0ef9e67f98b4-3_507_1085_933_383} A is the minimum point of the curve at \(( 3 , - 4 )\) and B is the point \(( 5,0 )\).
On separate diagrams on graph paper, draw the graphs of the following. In each case give the coordinates of the images of the points A and B .

1. \(\quad y = \mathrm { f } ( x ) + 2\),
2. \(y = \mathrm { f } ( x + 2 )\).

5 The point \(\mathrm { P } ( 5,4 )\) is on the curve \(y = \mathrm { f } ( x )\). State the coordinates of the image of P when the graph of \(y = \mathrm { f } ( x )\) is transformed to the graph of

1. \(y = \mathrm { f } ( x - 5 )\),
2. \(y = \mathrm { f } ( x ) + 7\).

3 The point \(\mathrm { P } ( 6,3 )\) lies on the curve \(y = \mathrm { f } ( x )\). State the coordinates of the image of P after the transformation which maps \(y = \mathrm { f } ( x )\) onto

1. \(y = 3 \mathrm { f } ( x )\),
2. \(y = \mathrm { f } ( 4 x )\).

8

1. The point \(\mathrm { P } ( 4 , - 2 )\) lies on the curve \(y = \mathrm { f } ( x )\). Find the coordinates of the image of P when the curve is transformed to \(y = \mathrm { f } ( 5 x )\).
2. Describe fully a single transformation which maps the curve \(y = \sin x ^ { \circ }\) onto the curve \(y = \sin ( x - 90 ) ^ { \circ }\).

4 The curve \(y = \mathrm { f } ( x )\) passes through the point \(P\) with coordinates \(( 2,5 )\).

1. State the coordinates of the point corresponding to \(P\) on the curve \(y = \mathrm { f } ( x ) + 2\).
2. State the coordinates of the point corresponding to \(P\) on the curve \(y = \mathrm { f } ( 2 x )\).
3. Describe the transformation that transforms the curve \(y = \mathrm { f } ( x )\) to the curve \(y = \mathrm { f } ( x + 4 )\).

4 The point \(\mathrm { P } ( 5,4 )\) is on the curve \(y = \mathrm { f } ( x )\). State the coordinates of the image of P when the graph of \(y = \mathrm { f } ( x )\) is transformed to the graph of

1. \(y = \mathrm { f } ( x - 5 )\),
2. \(y = \mathrm { f } ( x ) + 7\).

3

1. The point \(\mathrm { P } ( 4 , - 2 )\) lies on the curve \(y = \mathrm { f } ( x )\). Find the coordinates of the image of P when the curve is transformed to \(y = \mathrm { f } ( 5 x )\).
2. Describe fully a single transformation which maps the curve \(y = \sin x ^ { \circ }\) onto the curve \(y = \sin ( x - 90 ) ^ { \circ }\).

4 The point \(\mathrm { P } ( 6,3 )\) lies on the curve \(y = \mathrm { f } ( x )\). State the coordinates of the image of P after the transformation which maps \(y = \mathrm { f } ( x )\) onto

1. \(y = 3 \mathrm { f } ( x )\),
2. \(y = \mathrm { f } ( 4 x )\).

4 The point \(\mathrm { R } ( 6 , - 3 )\) is on the curve \(y = \mathrm { f } ( x )\).

1. Find the coordinates of the image of R when the curve is transformed to \(y = \frac { 1 } { 2 } \mathrm { f } ( x )\).
2. Find the coordinates of the image of R when the curve is transformed to \(y = \mathrm { f } ( 3 x )\).

9 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\), which has a \(y\)-intercept at \(\mathrm { P } ( 0,3 )\), a minimum point at \(\mathrm { Q } ( 1,2 )\), and an asymptote \(x = - 1\). \begin{figure}[h] \end{figure}

1. Find the coordinates of the images of the points P and Q when the curve \(y = \mathrm { f } ( x )\) is transformed to
   (A) \(y = 2 \mathrm { f } ( x )\),
   (B) \(y = \mathrm { f } ( x + 1 ) + 2\). You are now given that \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + 3 } { x + 1 } , x \neq - 1\).
2. Find \(\mathrm { f } ^ { \prime } ( x )\), and hence find the coordinates of the other turning point on the curve \(y = \mathrm { f } ( x )\).
3. Show that \(\mathrm { f } ( x - 1 ) = x - 2 + \frac { 4 } { x }\).
4. Find \(\int _ { a } ^ { b } \left( x - 2 + \frac { 4 } { x } \right) \mathrm { d } x\) in terms of \(a\) and \(b\). Hence, by choosing suitable values for \(a\) and \(b\), find the exact area enclosed by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis, the \(y\)-axis and the line \(x = 1\).

5

1. The graph of the function \(y = \mathrm { f } ( x )\) passes through the point \(P\) with coordinates (2, 6), and is a one-one function. State the coordinates of the point corresponding to \(P\) on each of the following curves.
   1. \(\quad y = \mathrm { f } ( x ) + 3\)
   2. \(\quad y = 2 \mathrm { f } ( 3 x - 1 )\)
   3. \(y = \mathrm { f } ^ { - 1 } ( x )\)
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{6b766f5c-8533-4e0c-bb10-0d9949dc777b-5_494_739_806_333} The diagram shows part of the graph of \(y = \mathrm { g } ^ { \prime } ( x )\). This is the graph of the gradient function of \(y = \mathrm { g } ( x )\). The graph intersects the \(x\)-axis at \(x = - 2\) and \(x = 4\).
   1. State the \(x\)-coordinate of any stationary points on the graph of \(y = \mathrm { g } ( x )\).
   2. State the set of values of \(x\) for which \(y = \mathrm { g } ( x )\) is a decreasing function.
   3. State the \(x\)-coordinate of any points of inflection on the graph of \(y = \mathrm { g } ( x )\).

1. The point \(P ( 2 , - 3 )\) lies on the curve with equation \(y = \mathrm { f } ( x )\).

State the coordinates of the image of \(P\) under the transformation represented by the curve
a. \(\quad y = | \mathrm { f } ( x ) |\)
b. \(y = \mathrm { f } ( x - 2 )\)
c. \(y = 3 \mathrm { f } ( 2 x ) + 2\)

1. The point \(P ( 3 , - 2 )\) lies on the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\)

Find the coordinates of the point to which \(P\) is mapped when the curve with equation \(y = \mathrm { f } ( x )\) is transformed to the curve with equation

1. \(y = \mathrm { f } ( x - 2 )\)
2. \(y = \mathrm { f } ( 2 x )\)
3. \(y = 3 \mathrm { f } ( - x ) + 5\)