Multiple separate transformations (sketch-based, standard transformations)

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\).
The curve passes through the points \(( - 1,0 )\) and \(( 0,2 )\) and touches the \(x\)-axis at the point \(( 3,0 )\). On separate diagrams, sketch the curve with equation

1. \(y = \mathrm { f } ( \mathrm { x } + 3 )\)
2. \(y = \mathrm { f } ( - 3 x )\) On each diagram, show clearly the coordinates of all the points where the curve cuts or touches the coordinate axes.

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the graph of \(y = \mathrm { f } ( x )\).
The graph intersects the \(y\)-axis at the point \(( 0,1 )\) and the point \(A ( 2,3 )\) is the maximum turning point. Sketch, on separate axes, the graphs of

1. \(y = \mathrm { f } ( - x ) + 1\),
2. \(y = \mathrm { f } ( x + 2 ) + 3\),
3. \(y = 2 \mathrm { f } ( 2 x )\). On each sketch, show the coordinates of the point at which your graph intersects the \(y\)-axis and the coordinates of the point to which \(A\) is transformed.

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

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve has a maximum at \(( - 3,4 )\) and a minimum at \(( 1 , - 2 )\). Showing the coordinates of any turning points, sketch on separate diagrams the curves with equations

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

5 Answer this question on the insert provided. Fig. 5 shows the graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h] \end{figure} On the insert, draw the graph of

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

9 Answer parts (i) and (iii) on the insert provided. Fig. 9 shows a sketch graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h] \end{figure}

1. On the Insert sketch graphs of
   (A) \(y = 2 \mathrm { f } ( x )\),
   (B) \(y = \mathrm { f } ( - x )\),
   (C) \(y = \mathrm { f } ( x - 2 )\) In each case describe the transformations.
2. Explain why the function \(y = \mathrm { f } ( x )\) does not have an inverse function.
3. The function \(\mathrm { g } ( x )\) is defined as follows: $$\mathrm { g } ( x ) = \mathrm { f } ( x ) \text { for } x \geq 0$$ On the Insert sketch the graph of \(y = \mathrm { g } ^ { - 1 } ( x )\).
4. You are given that \(\mathrm { f } ( x ) = x ^ { 2 } ( x + 2 )\). Calculate the gradient of the curve \(y = \mathrm { f } ( x )\) at the point \(( 1,3 )\).
   Deduce the gradient of the function \(\mathrm { g } ^ { - 1 } ( x )\) at the point where \(x = 3\).
5. Show that \(\mathrm { g } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) cross where \(x = - 1 + \sqrt { 2 }\). \section*{Insert for question 9.}
6. (A) On the axes below sketch the graph of \(y = 2 \mathrm { f } ( x )\). Describe the transformation. \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-5_563_1102_484_395} Description:
7. (B) On the axes below sketch the graph of \(y = \mathrm { f } ( - x )\). Describe the transformation. \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-5_588_1154_1576_404} Description:
8. (C) On the axes below sketch the graph of \(y = \mathrm { f } ( x - 2 )\). Describe the transformation. \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-6_615_1230_402_406} Description:
9. The function \(\mathrm { g } ( x )\) is defined as follows: $$\mathrm { g } ( x ) = \mathrm { f } ( x ) \text { for } x \geq 0$$ On the axes below sketch the graph of \(y = g ^ { - 1 } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-6_677_1356_1567_312}

4 The graph of \(\mathrm { f } ( x )\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{69792771-6de6-4886-9c71-e794fcb7aaba-2_949_1127_1041_507} Draw the graphs of

1. \(\mathrm { f } ( x + 2 ) + 1\),
2. \(- \frac { 1 } { 2 } \mathrm { f } ( x )\).

4 The graph of \(\mathrm { f } ( x )\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{29e924de-bedf-4719-bbfe-f5e0d3191d59-2_949_1127_1041_507} Draw the graphs of

1. \(\mathrm { f } ( x + 2 ) + 1\),
2. \(- \frac { 1 } { 2 } \mathrm { f } ( x )\).

\includegraphics{figure_1} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \text{f}(x)\) 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. \(y = \text{f}(2x)\) [3]
2. \(y = \text{f}(x + p)\), where \(p\) is a constant and \(0 < p < 3\) [4]

On each diagram show the coordinates of any points where the curve intersects the \(x\)-axis and of any minimum or maximum points.

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = f(x)\). The curve crosses the \(x\)-axis at the points \((2, 0)\) and \((4, 0)\). The minimum point on the curve is \(P(3, -2)\). In separate diagrams sketch the curve with equation

1. \(y = -f(x)\), [3]
2. \(y = f(2x)\). [3]

On each diagram, give the coordinates of the points at which the curve crosses the \(x\)-axis, and the coordinates of the image of \(P\) under the given transformation.

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = f(x)\). The curve passes through the origin \(O\) and through the point \((6, 0)\). The maximum point on the curve is \((3, 5)\). On separate diagrams, sketch the curve with equation

1. \(y = 3f(x)\), [2]
2. \(y = f(x + 2)\). [3]

On each diagram, show clearly the coordinates of the maximum point and of each point at which the curve crosses the \(x\)-axis.

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = f(x)\). The curve passes through the points \((0, 3)\) and \((4, 0)\) and touches the \(x\)-axis at the point \((1, 0)\). On separate diagrams, sketch the curve with equation

1. \(y = f(x + 1)\), [3]
2. \(y = 2f(x)\), [3]
3. \(y = f\left(\frac{1}{2}x\right)\). [3]

On each diagram show clearly the coordinates of all the points at which the curve meets the axes.

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = \text{f}(x)\). The curve crosses the coordinate axes at the points \((0, 1)\) and \((3, 0)\). The maximum point on the curve is \((1, 2)\). On separate diagrams, sketch the curve with equation

1. \(y = \text{f}(x + 1)\), [3]
2. \(y = \text{f}(2x)\). [3]

On each diagram, show clearly the coordinates of the maximum point, and of each point at which the curve crosses the coordinate axes.

\includegraphics{figure_1} Figure 1 shows a sketch of a curve with equation \(y = f(x)\). The curve crosses the \(y\)-axis at \((0, 3)\) and has a minimum at \(P(4, 2)\). On separate diagrams, sketch the curve with equation

1. \(y = f(x + 4)\), [2]
2. \(y = 2f(x)\). [2]

On each diagram, show clearly the coordinates of the minimum point and any point of intersection with the \(y\)-axis.

\includegraphics{figure_5} The diagram shows a sketch of the curve with equation \(y = f(x)\). The curve has a maximum at \((-3, 4)\) and a minimum at \((1, -2)\). Showing the coordinates of any turning points, sketch on separate diagrams the curves with equations

1. \(y = 2f(x)\), [3]
2. \(y = -f(x)\). [3]

\includegraphics{figure_5} Fig. 5 shows a sketch of the graph of \(y = f(x)\). On separate diagrams, sketch the graphs of the following, showing clearly the coordinates of the points corresponding to P, Q and R.

1. \(y = f(2x)\) [2]
2. \(y = \frac{1}{2}f(x)\) [2]

In this question, \(f(x) = x^2 - 5x\). Fig. 4 shows a sketch of the graph of \(y = f(x)\). \includegraphics{figure_4} On separate diagrams, sketch the curves \(y = f(2x)\) and \(y = 3f(x)\), labelling the coordinates of their intersections with the axes and their turning points. [4]

Fig. 8 shows the graph of \(y = g(x)\). \includegraphics{figure_8} Draw the graph of

1. \(y = g(2x)\), [2]
2. \(y = 3g(x)\). [2]

\includegraphics{figure_3} The diagram shows the graph of \(y = \text{g}(x)\). In the printed answer booklet, using the same scale as in this diagram, sketch the curves

1. \(y = \frac{3}{2}\text{g}(x)\), [2]
2. \(y = \text{g}\left(\frac{1}{2}x\right)\). [2]