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

9. $$\mathrm { f } ( \theta ) = 5 \cos \theta - 4 \sin \theta \quad \theta \in \mathbb { R }$$

1. Express \(\mathrm { f } ( \theta )\) in the form \(R \cos ( \theta + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the exact value of \(R\) and give the value of \(\alpha\), in radians, to 3 decimal places. The curve with equation \(y = \cos \theta\) is transformed onto the curve with equation \(y = \mathrm { f } ( \theta )\) by a sequence of two transformations. Given that the first transformation is a stretch and the second a translation,
2. 1. describe fully the transformation that is a stretch,
   2. describe fully the transformation that is a translation. Given $$g ( \theta ) = \frac { 90 } { 4 + ( f ( \theta ) ) ^ { 2 } } \quad \theta \in \mathbb { R }$$
3. find the range of g.

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2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where \(x \in \mathbb { R }\) and \(\mathrm { f } ( x )\) is a polynomial. The curve passes through the origin and touches the \(x\)-axis at the point \(( 3,0 )\) There is a maximum turning point at \(( 1,2 )\) and a minimum turning point at \(( 3,0 )\) On separate diagrams, sketch the curve with equation

1. \(y = 3 f ( 2 x )\)
2. \(y = \mathrm { f } ( - x ) - 1\) On each sketch, show clearly the coordinates of
   • the point where the curve crosses the \(y\)-axis
   • any maximum or minimum turning points

7. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the graph with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = \frac { 1 } { 2 } | 2 x + 7 | - 10$$

1. State the coordinates of the vertex, V, of the graph.
2. Solve, using algebra, $$\frac { 1 } { 2 } | 2 x + 7 | - 10 \geqslant \frac { 1 } { 3 } x + 1$$
3. Sketch the graph with equation $$y = | \mathrm { f } ( x ) |$$ stating the coordinates of the local maximum point and each local minimum point.

2. $$f ( x ) = \cos x + 2 \sin x$$

1. Express \(\mathrm { f } ( x )\) in the form \(R \cos ( x - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and give the value of \(\alpha\), in radians, to 3 decimal places. $$g ( x ) = 3 - 7 f ( 2 x )$$
2. Using the answer to part (a),
   1. write down the exact maximum value of \(\mathrm { g } ( x )\),
   2. find the smallest positive value of \(x\) for which this maximum value occurs, giving your answer to 2 decimal places.

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. \(y = | f ( x ) |\)

8. \begin{figure}[h] \end{figure} \section*{In this question you must show all stages of your working.} \section*{Solutions relying on calculator technology are not acceptable.} The graph shown in Figure 2 has equation $$y = a - | 2 x - b |$$ where \(a\) and \(b\) are positive constants, \(a > b\)

1. Find, giving your answer in terms of \(a\) and \(b\),
   1. the coordinates of the maximum point of the graph,
   2. the coordinates of the point of intersection of the graph with the \(y\)-axis,
   3. the coordinates of the points of intersection of the graph with the \(x\)-axis. On page 24 there is a copy of Figure 2 called Diagram 1.
2. On Diagram 1, sketch the graph with equation $$y = | x | - 1$$ Given that the graphs \(y = | x | - 1\) and \(y = a - | 2 x - b |\) intersect at \(x = - 3\) and \(x = 5\)
3. find the value of \(a\) and the value of \(b\) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{76989f19-2624-4e86-a8ee-4978dd1014c2-24_675_652_1959_712} \captionsetup{labelformat=empty} \caption{Diagram 1}
   \end{figure}

8. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( 2 x + 1 ) ^ { 3 } e ^ { - 4 x }$$

1. Show that $$\mathrm { f } ^ { \prime } ( x ) = A ( 2 x + 1 ) ^ { 2 } ( 1 - 4 x ) \mathrm { e } ^ { - 4 x }$$ where \(A\) is a constant to be found.
2. Hence find the exact coordinates of the two stationary points on \(C\). The function g is defined by $$g ( x ) = 8 f ( x - 2 )$$
3. Find the coordinates of the maximum stationary point on the curve with equation \(y = g ( x )\).

1. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the graph with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = 2 | x - 5 | + 10$$ The point \(P\), shown in Figure 1, is the vertex of the graph.

1. State the coordinates of \(P\)
2. Use algebra to solve $$2 | x - 5 | + 10 > 6 x$$ (Solutions relying on calculator technology are not acceptable.)
3. Find the point to which \(P\) is mapped, when the graph with equation \(y = \mathrm { f } ( x )\) is transformed to the graph with equation \(y = 3 \mathrm { f } ( x - 2 )\)

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the graph with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = 21 - 2 | 2 - x | \quad x \geqslant 0$$

1. Find ff(6)
2. Solve the equation \(\mathrm { f } ( x ) = 5 x\) Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has exactly two roots,
3. state the set of possible values of \(k\). The graph with equation \(y = \mathrm { f } ( x )\) is transformed onto the graph with equation \(y = a \mathrm { f } ( x - b )\) The vertex of the graph with equation \(y = a \mathrm { f } ( x - b )\) is (6, 3). Given that \(a\) and \(b\) are constants,
4. find the value of \(a\) and the value of \(b\). \includegraphics[max width=\textwidth, alt={}, center]{96948fd3-5438-4e95-b41b-2f649ca8dfac-11_2255_50_314_34}

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the graph with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = | 3 x - 13 | + 5 \quad x \in \mathbb { R }$$ The vertex of the graph is at point \(P\), as shown in Figure 1.

1. State the coordinates of \(P\).
2. 1. State the range of f .
   2. Find the value of ff(4)
3. Solve, using algebra and showing your working, $$16 - 2 x > | 3 x - 13 | + 5$$ The graph with equation \(y = \mathrm { f } ( x )\) is transformed onto the graph with equation \(y = a \mathrm { f } ( x + b )\) The vertex of the graph with equation \(y = a \mathrm { f } ( x + b )\) is \(( 4,20 )\) Given that \(a\) and \(b\) are constants,
4. find the value of \(a\) and the value of \(b\).

10. (a) Express \(3 \sin 2 x + 5 \cos 2 x\) in the form \(R \sin ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and give the value of \(\alpha\) to 3 significant figures.
(b) Solve, for \(0 < x < \pi\), $$3 \sin 2 x + 5 \cos 2 x = 4$$ (Solutions based entirely on graphical or numerical methods are not acceptable.) $$g ( x ) = 4 ( 3 \sin 2 x + 5 \cos 2 x ) ^ { 2 } + 3$$ (c) Using your answer to part (a) and showing your working,

1. find the greatest value of \(\mathrm { g } ( x )\),
2. find the least value of \(\mathrm { g } ( x )\).

11. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x ) , \quad x \in \mathbb { R }\) The curve meets the coordinate axes at the points \(A ( 0 , - 3 )\) and \(B \left( - \frac { 1 } { 3 } \ln 4,0 \right)\) and the curve has an asymptote with equation \(y = - 4\) In separate diagrams, sketch the graph with equation

1. \(y = | f ( x ) |\)
2. \(y = 2 \mathrm { f } ( x ) + 6\) On each sketch, give the exact coordinates of the points where the curve crosses or meets the coordinate axes and the equation of any asymptote. Given that $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { - 3 x } - 4 , & x \in \mathbb { R } \\ \mathrm {~g} ( x ) = \ln \left( \frac { 1 } { x + 2 } \right) , & x > - 2 \end{array}$$
3. state the range of f,
4. find \(\mathrm { f } ^ { - 1 } ( x )\),
5. express \(f g ( x )\) as a polynomial in \(x\).

7. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the graph of \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\).
The point \(P \left( \frac { 1 } { 3 } , 0 \right)\) is the vertex of the graph.
The point \(Q ( 0,5 )\) is the intercept with the \(y\)-axis. Given that \(\mathrm { f } ( x ) = | a x + b |\), where \(a\) and \(b\) are constants,

1. 1. find all possible values for \(a\) and \(b\),
   2. hence find an equation for the graph.
2. Sketch the graph with equation $$y = \mathrm { f } \left( \frac { 1 } { 2 } x \right) + 3$$ showing the coordinates of its vertex and its intercept with the \(y\)-axis.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the graph with equation \(y = \mathrm { f } ( x ) , \quad x \in \mathbb { R }\) The graph consists of two half lines that meet at the point \(P ( 2 , - 3 )\), the vertex of the graph.
The graph cuts the \(y\)-axis at the point \(( 0 , - 1 )\) and the \(x\)-axis at the points \(( - 1,0 )\) and \(( 5,0 )\).
Sketch, on separate diagrams, the graph of

1. \(y = \mathrm { f } ( | x | )\),
2. \(y = 2 \mathrm { f } ( x + 5 )\). In each case, give the coordinates of the points where the graph crosses or meets the coordinate axes. Also give the coordinates of any vertices corresponding to the point \(P\).

11. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = \mathrm { e } ^ { a - 3 x } - 3 \mathrm { e } ^ { - x } , \quad x \in \mathbb { R }$$ where \(a\) is a constant and \(a > \ln 4\) The curve \(C\) has a turning point \(P\) and crosses the \(x\)-axis at the point \(Q\) as shown in Figure 2.

1. Find, in terms of \(a\), the coordinates of the point \(P\).
2. Find, in terms of \(a\), the \(x\) coordinate of the point \(Q\).
3. Sketch the curve with equation $$y = \left| \mathrm { e } ^ { a - 3 x } - 3 \mathrm { e } ^ { - x } \right| , \quad x \in \mathbb { R } , \quad a > \ln 4$$ Show on your sketch the exact coordinates, in terms of \(a\), of the points at which the curve meets or cuts the coordinate axes.

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \frac { 6 x + 2 } { 3 x ^ { 2 } + 5 } , \quad x \in \mathbb { R }$$

1. Find \(\mathrm { f } ^ { \prime } ( x )\), writing your answer as a single fraction in its simplest form. The curve has two turning points, a maximum at point \(A\) and a minimum at point \(B\), as shown in Figure 2.
2. Using part (a), find the coordinates of point \(A\) and the coordinates of point \(B\).
3. State the coordinates of the maximum turning point of the function with equation $$y = \mathrm { f } ( 2 x ) + 4 \quad x \in \mathbb { R }$$
4. Find the range of the function $$\operatorname { g } ( x ) = \frac { 6 x + 2 } { 3 x ^ { 2 } + 5 } , \quad x \leqslant 0$$

10. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the graph with equation \(y = \mathrm { g } ( x )\), where $$\mathrm { g } ( x ) = \frac { 3 x - 4 } { x - 3 } , \quad x \in \mathbb { R } , \quad x < 3$$ The graph cuts the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\), as shown in Figure 2 .

1. State the range of g .
2. State the coordinates of
   1. point \(A\)
   2. point \(B\)
3. Find \(\operatorname { gg } ( x )\) in its simplest form.
4. Sketch the graph with equation \(y = | \mathrm { g } ( x ) |\) On your sketch, show the coordinates of each point at which the graph meets or cuts the axes and state the equation of each asymptote.
5. Find the exact solution of the equation \(| \mathrm { g } ( x ) | = 8\)

1. The functions \(f\) and \(g\) are defined by

$$\begin{array} { l l } \mathrm { f } : x \mapsto \mathrm { e } ^ { - x } + 2 , & x \in \mathbb { R } \\ \mathrm {~g} : x \mapsto 2 \ln x , & x > 0 \end{array}$$

1. Find \(\mathrm { fg } ( x )\), giving your answer in its simplest form.
2. Find the exact value of \(x\) for which \(\mathrm { f } ( 2 x + 3 ) = 6\)
3. Find \(\mathrm { f } ^ { - 1 }\), stating its domain.
4. On the same axes, sketch the curves with equation \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), giving the coordinates of all the points where the curves cross the axes.

1. \begin{figure}[h] \end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x ) , - 5 \leqslant x \leqslant 5\).
The point \(M ( 2,4 )\) is the maximum turning point of the graph.
Sketch, on separate diagrams, the graphs of

1. \(y = \mathrm { f } ( x ) + 3\),
2. \(y = | \mathrm { f } ( x ) |\),
3. \(y = \mathrm { f } ( | x | )\). Show on each graph the coordinates of any maximum turning points.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\).
The curve passes through the origin \(O\) and the points \(A ( 5,4 )\) and \(B ( - 5 , - 4 )\).
In separate diagrams, sketch the graph with equation

1. \(y = | f ( x ) |\),
2. \(y = \mathrm { f } ( | x | )\),
3. \(y = 2 f ( x + 1 )\). On each sketch, show the coordinates of the points corresponding to \(A\) and \(B\).

3. \begin{figure}[h] \end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x ) , \quad 1 < x < 9\).
The points \(T ( 3,5 )\) and \(S ( 7,2 )\) are turning points on the graph.
Sketch, on separate diagrams, the graphs of

1. \(y = 2 \mathrm { f } ( x ) - 4\),
2. \(y = | \mathrm { f } ( x ) |\). Indicate on each diagram the coordinates of any turning points on your sketch.

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.

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

1. Find the value of ff(-3). On separate diagrams, sketch the curve with equation
2. \(y = \mathrm { f } ^ { - 1 } ( x )\),
3. \(y = \mathrm { f } ( | x | ) - 2\),
4. \(y = 2 \mathrm { f } \left( \frac { 1 } { 2 } x \right)\). Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.

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 \(\mathrm { f } ( x ) = 5 x\).

3. \begin{figure}[h] \end{figure} Figure 1 shows the graph of \(y = f ( x ) , x \in \mathbb { R }\).
The graph consists of two line segments that meet at the point \(P\).
The graph cuts the \(y\)-axis at the point \(Q\) and the \(x\)-axis at the points \(( - 3,0 )\) and \(R\). Sketch, on separate diagrams, the graphs of

1. \(y = | f ( x ) |\),
2. \(y = \mathrm { f } ( - x )\). Given that \(\mathrm { f } ( x ) = 2 - | x + 1 |\),
3. find the coordinates of the points \(P , Q\) and \(R\),
4. solve \(\mathrm { f } ( x ) = \frac { 1 } { 2 } x\).