1.02w Graph transformations: simple transformations of f(x)

5. Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\).
The curve meets the coordinate axes at the points \(A ( 0,1 - k )\) and \(B \left( \frac { 1 } { 2 } \ln k , 0 \right)\), where \(k\) is a constant and \(k > 1\), as shown in Figure 2. On separate diagrams, sketch the curve with equation

1. \(y = | f ( x ) |\),
2. \(y = \mathrm { f } ^ { - 1 } ( x )\). Show on each sketch the coordinates, in terms of \(k\), of each point at which the curve meets or cuts the axes. Given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } - k\),
3. state the range of f ,
4. find \(\mathrm { f } ^ { - 1 } ( x )\),
5. write down the domain of \(\mathrm { f } ^ { - 1 }\).

6. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with the equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The curve has a turning point at \(A ( 3 , - 4 )\) and also passes through the point \(( 0,5 )\).

1. Write down the coordinates of the point to which \(A\) is transformed on the curve with equation
   1. \(y = | \mathrm { f } ( x ) |\),
   2. \(y = 2 f \left( \frac { 1 } { 2 } x \right)\).
2. Sketch the curve with equation $$y = \mathrm { f } ( | x | )$$ On your sketch show the coordinates of all turning points and the coordinates of the point at which the curve cuts the \(y\)-axis. The curve with equation \(y = \mathrm { f } ( x )\) is a translation of the curve with equation \(y = x ^ { 2 }\).
3. Find \(\mathrm { f } ( x )\).
4. Explain why the function f does not have an inverse.

3. \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 \(R ( 4 , - 3 )\), as shown in Figure 1. Sketch, on separate diagrams, the graphs of

1. \(y = 2 \mathrm { f } ( x + 4 )\),
2. \(y = | \mathrm { f } ( - x ) |\). On each diagram, show the coordinates of the point corresponding to \(R\).

4. \begin{figure}[h] \end{figure} Figure 2 shows part of the curve with equation \(y = \mathrm { f } ( x )\) The curve passes through the points \(P ( - 1.5,0 )\) and \(Q ( 0,5 )\) as shown.
On separate diagrams, sketch the curve with equation

1. \(y = | f ( x ) |\)
2. \(y = \mathrm { f } ( | x | )\)
3. \(y = 2 f ( 3 x )\) Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x ) , x > 0\), where f is an increasing function of \(x\). The curve crosses the \(x\)-axis at the point \(( 1,0 )\) and the line \(x = 0\) is an asymptote to the curve. On separate diagrams, sketch the curve with equation

1. \(y = \mathrm { f } ( 2 x ) , x > 0\)
2. \(y = | \mathrm { f } ( x ) | , x > 0\) Indicate clearly on each sketch the coordinates of the point at which the curve crosses or meets the \(x\)-axis.

1. $$g ( x ) = \frac { 6 x + 12 } { x ^ { 2 } + 3 x + 2 } - 2 , \quad x \geqslant 0$$

1. Show that \(\mathrm { g } ( x ) = \frac { 4 - 2 x } { x + 1 } , x \geqslant 0\)
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0f6fd881-4d4b-4f80-96cc-6da41cc33c60-02_494_922_628_511} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { g } ( x ) , x \geqslant 0\) The curve meets the \(y\)-axis at \(( 0,4 )\) and crosses the \(x\)-axis at \(( 2,0 )\). On separate diagrams sketch the graph with equation
   1. \(y = 2 \mathrm {~g} ( 2 x )\),
   2. \(y = \mathrm { g } ^ { - 1 } ( x )\). Show on each sketch the coordinates of each point at which the graph meets or crosses the axes.

2. Given that $$\mathrm { f } ( x ) = \ln x , \quad x > 0$$ sketch on separate axes the graphs of

1. \(\quad y = \mathrm { f } ( x )\),
2. \(y = | \mathrm { f } ( x ) |\),
3. \(y = - \mathrm { f } ( x - 4 )\). Show, on each diagram, the point where the graph meets or crosses the \(x\)-axis. In each case, state the equation of the asymptote.

4. \begin{figure}[h] \end{figure} Figure 1 shows part of the graph with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The graph consists of two line segments that meet at the point \(Q ( 6 , - 1 )\). The graph crosses the \(y\)-axis at the point \(P ( 0,11 )\). Sketch, on separate diagrams, the graphs of

1. \(y = | f ( x ) |\)
2. \(y = 2 f ( - x ) + 3\) On each diagram, show the coordinates of the points corresponding to \(P\) and \(Q\).
   Given that \(\mathrm { f } ( x ) = a | x - b | - 1\), where \(a\) and \(b\) are constants,
3. state the value of \(a\) and the value of \(b\).

5. \begin{figure}[h] \end{figure} Figure 2 shows part of the graph with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 2 | 5 - x | + 3 , \quad x \geqslant 0$$ Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has exactly one root,

1. state the set of possible values of \(k\).
2. Solve the equation \(\mathrm { f } ( x ) = \frac { 1 } { 2 } x + 10\) The graph with equation \(y = \mathrm { f } ( x )\) is transformed onto the graph with equation \(y = 4 \mathrm { f } ( x - 1 )\). The vertex on the graph with equation \(y = 4 \mathrm { f } ( x - 1 )\) has coordinates \(( p , q )\).
3. State the value of \(p\) and the value of \(q\).

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The curve has a minimum point at \(( - 0.5 , - 2 )\) and a maximum point at \(( 0.4 , - 4 )\). The lines \(x = 1\), the \(x\)-axis and the \(y\)-axis are asymptotes of the curve, as shown in Fig. 1. On a separate diagram sketch the graphs of

1. \(y = | \mathrm { f } ( x ) |\),
2. \(y = \mathrm { f } ( x - 3 )\),
3. \(y = \mathrm { f } ( | x | )\). In each case show clearly
   1. the coordinates of any points at which the curve has a maximum or minimum point,
   2. how the curve approaches the asymptotes of the curve.
      6. continued

5. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The curve meets the \(x\)-axis at \(P ( p , 0 )\) and meets the \(y\)-axis at \(Q ( 0 , q )\).

1. On separate diagrams, sketch the curve with equation
   1. \(y = | \mathrm { f } ( x ) |\),
   2. \(y = 3 \mathrm { f } \left( \frac { 1 } { 2 } x \right)\). In each case show, in terms of \(p\) or \(q\), the coordinates of points at which the curve meets the axes. Given that \(\mathrm { f } ( x ) = 3 \ln ( 2 x + 3 )\),
2. state the exact value of \(q\),
3. find the value of \(p\),
4. find an equation for the tangent to the curve at \(P\).

\begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
The curve crosses the coordinate axes at the points \(( - 6,0 )\) and \(( 0,3 )\), has a stationary point at \(( - 3,9 )\) and has an asymptote with equation \(y = 1\) On separate diagrams, sketch the curve with equation

1. \(y = - \mathrm { f } ( x )\)
2. \(y = \mathrm { f } \left( \frac { 3 } { 2 } x \right)\) On each diagram, show clearly the coordinates of the points of intersection of the curve with the two coordinate axes, the coordinates of the stationary point, and the equation of the asymptote. \includegraphics[max width=\textwidth, alt={}, center]{de511cb3-35c7-4225-b459-a136b6304b78-07_2255_45_316_36}

\begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \frac { 3 } { x } , x \neq 0\).

1. On a separate diagram, sketch the curve with equation \(y = \frac { 3 } { x + 2 } , x \neq - 2\), showing the coordinates of any point at which the curve crosses a coordinate axis.
2. Write down the equations of the asymptotes of the curve in part (a).

Figure 1 shows the graph of equation \(y = \mathrm { f } ( x )\).
The points \(P ( - 3,0 )\) and \(Q ( 2 , - 4 )\) are stationary points on the graph.
Sketch, on separate diagrams, the graphs of

1. \(y = 3 \mathrm { f } ( x + 2 )\)
2. \(y = | \mathrm { f } ( x ) |\) On each diagram, show the coordinates of any stationary points.

\begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\), where f is an increasing function of \(x\). The curve passes through the points \(P ( 0 , - 2 )\) and \(Q ( 3,0 )\) as shown. In separate diagrams, sketch the curve with equation

1. \(y = | f ( x ) |\),
2. \(y = \mathrm { f } ^ { - 1 } ( x )\),
3. \(y = \frac { 1 } { 2 } \mathrm { f } ( 3 x )\). Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.

5 A curve has parametric equations \(x = \lambda \cos \theta - \frac { 1 } { \lambda } \sin \theta , y = \cos \theta + \sin \theta\), where \(\lambda\) is a positive constant.

1. Use your calculator to obtain a sketch of the curve in each of the cases $$\lambda = 0.5 , \quad \lambda = 3 \quad \text { and } \quad \lambda = 5 .$$
2. Given that the curve is a conic, name the type of conic.
3. Show that \(y\) has a maximum value of \(\sqrt { 2 }\) when \(\theta = \frac { 1 } { 4 } \pi\).
4. Show that \(x ^ { 2 } + y ^ { 2 } = \left( 1 + \lambda ^ { 2 } \right) + \left( \frac { 1 } { \lambda ^ { 2 } } - \lambda ^ { 2 } \right) \sin ^ { 2 } \theta\), and deduce that the distance from the origin of any point on the curve is between \(\sqrt { 1 + \frac { 1 } { \lambda ^ { 2 } } }\) and \(\sqrt { 1 + \lambda ^ { 2 } }\).
5. For the case \(\lambda = 1\), show that the curve is a circle, and find its radius.
6. For the case \(\lambda = 2\), draw a sketch of the curve, and label the points \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } , \mathrm { G } , \mathrm { H }\) on the curve corresponding to \(\theta = 0 , \frac { 1 } { 4 } \pi , \frac { 1 } { 2 } \pi , \frac { 3 } { 4 } \pi , \pi , \frac { 5 } { 4 } \pi , \frac { 3 } { 2 } \pi , \frac { 7 } { 4 } \pi\) respectively. You should make clear what is special about each of these points.

5 In parts (i), (ii), (iii) of this question you are required to investigate curves with the equation $$x ^ { k } + y ^ { k } = 1$$ for various positive values of \(k\).

1. Firstly consider cases in which \(k\) is a positive even integer.
   (A) State the shape of the curve when \(k = 2\).
   (B) Sketch, on the same axes, the curves for \(k = 2\) and \(k = 4\).
   (C) Describe the shape that the curve tends to as \(k\) becomes very large.
   (D) State the range of possible values of \(x\) and \(y\).
2. Now consider cases in which \(k\) is a positive odd integer.
   (A) Explain why \(x\) and \(y\) may take any value.
   (B) State the shape of the curve when \(k = 1\).
   (C) Sketch the curve for \(k = 3\). State the equation of the asymptote of this curve.
   (D) Sketch the shape that the curve tends to as \(k\) becomes very large.
3. Now let \(k = \frac { 1 } { 2 }\). Sketch the curve, indicating the range of possible values of \(x\) and \(y\).
4. Now consider the modified equation \(| x | ^ { k } + | y | ^ { k } = 1\).
   (A) Sketch the curve for \(k = \frac { 1 } { 2 }\).
   (B) Investigate the shape of the curve for \(k = \frac { 1 } { n }\) as the positive integer \(n\) becomes very large.

3

1. The curve \(y = 5 \sqrt { } x\) is transformed by a stretch, scale factor \(\frac { 1 } { 2 }\), parallel to the \(x\)-axis. Find the equation of the curve after it has been transformed.
2. Describe the single transformation which transforms the curve \(y = 5 \sqrt { } x\) to the curve \(y = ( 5 \sqrt { } x ) - 3\).

4

1. Sketch the curve \(y = \frac { 1 } { x ^ { 2 } }\).
2. Hence sketch the curve \(y = \frac { 1 } { ( x - 3 ) ^ { 2 } }\).
3. Describe fully a transformation that transforms the curve \(y = \frac { 1 } { x ^ { 2 } }\) to the curve \(y = \frac { 2 } { x ^ { 2 } }\).

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

5

1. Sketch the curve \(y = x ^ { 3 } + 2\).
2. Sketch the curve \(y = 2 \sqrt { x }\).
3. Describe a transformation that transforms the curve \(y = 2 \sqrt { x }\) to the curve \(y = 3 \sqrt { x }\).

3

1. Sketch the curve \(y = x ^ { 3 }\).
2. Describe a transformation that transforms the curve \(y = x ^ { 3 }\) to the curve \(y = - x ^ { 3 }\).
3. The curve \(y = x ^ { 3 }\) is translated by \(p\) units, parallel to the \(x\)-axis. State the equation of the curve after it has been transformed.

2

1. On separate diagrams, sketch the graphs of
   1. \(\mathrm { y } = \frac { 1 } { \mathrm { x } }\),
   2. \(y = x ^ { 4 }\).
2. Describe a transformation that transforms the curve \(y = x ^ { 3 }\) to the curve \(y = 8 x ^ { 3 }\).

2

1. The curve \(y = x ^ { 2 }\) is translated 2 units in the positive \(x\)-direction. Find the equation of the curve after it has been translated.
2. The curve \(y = x ^ { 3 } - 4\) is reflected in the \(x\)-axis. Find the equation of the curve after it has been reflected.

6

1. Sketch the graph of \(y = \frac { 1 } { x }\), where \(x \neq 0\), showing the parts of the graph corresponding to both positive and negative values of \(x\).
2. Describe fully the geometrical transformation that transforms the curve \(y = \frac { 1 } { x }\) to the curve \(y = \frac { 1 } { x + 2 }\). Hence sketch the curve \(y = \frac { 1 } { x + 2 }\).
3. Differentiate \(\frac { 1 } { x }\) with respect to \(x\).
4. Use parts (ii) and (iii) to find the gradient of the curve \(y = \frac { 1 } { x + 2 }\) at the point where it crosses the \(y\)-axis.