1.02x Combinations of transformations: multiple transformations

7 The diagram shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-06_620_1216_356_422}

1. On Figure 1, below, sketch the curve with equation \(y = - \mathrm { f } ( 3 x )\), indicating the values where the curve cuts the coordinate axes.
2. On Figure 2, on page 7, sketch the curve with equation \(y = \mathrm { f } ( | x | )\), indicating the values where the curve cuts the coordinate axes.
3. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { f } \left( - \frac { 1 } { 2 } x \right)\). \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{063bbfa5-df49-44a1-8143-5e076397f63f-06_732_1237_1649_443}
   \end{figure} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{063bbfa5-df49-44a1-8143-5e076397f63f-07_727_1211_340_466}
   \end{figure}

4 The sketch shows part of the curve with equation \(y = \mathrm { f } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-08_536_1054_367_539}

1. On Figure 2 below, sketch the curve with equation \(y = - | \mathrm { f } ( x ) |\).
2. On Figure 3 on the page opposite, sketch the curve with equation \(y = \mathrm { f } ( | 2 x | )\).
3. 1. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { f } ( 2 x + 2 )\).
   2. Find the coordinates of the image of the point \(P ( 4 , - 3 )\) under the sequence of transformations given in part (c)(i). \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-09_778_1032_424_529}
      \end{figure}

5. \begin{figure}[h] \end{figure} Figure 1 shows the curve \(y = \mathrm { f } ( x )\) which has a maximum point at ( \(- 3,2\) ) and a minimum point at \(( 2 , - 4 )\).

1. Showing the coordinates of any stationary points, sketch on separate diagrams the graphs of
   1. \(y = \mathrm { f } ( | x | )\),
   2. \(y = 3 \mathrm { f } ( 2 x )\).
2. Write down the values of the constants \(a\) and \(b\) such that the curve with equation \(y = a + \mathrm { f } ( x + b )\) has a minimum point at the origin \(O\).

6. \begin{figure}[h] \end{figure} Figure 1 shows the curve \(y = \mathrm { f } ( x )\) which has a minimum point at \(\left( - \frac { 3 } { 2 } , 0 \right)\), a maximum point at \(( 3,6 )\) and crosses the \(y\)-axis at \(( 0,4 )\). Sketch each of the following graphs on separate diagrams. In each case, show the coordinates of any turning points and of any points where the graph meets the coordinate axes.

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

7. \begin{figure}[h] \end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x )\) which meets the coordinate axes at the points \(( a , 0 )\) and \(( 0 , b )\), where \(a\) and \(b\) are constants.

1. Showing, in terms of \(a\) and \(b\), the coordinates of any points of intersection with the axes, sketch on separate diagrams the graphs of
   1. \(\quad y = \mathrm { f } ^ { - 1 } ( x )\),
   2. \(y = 2 \mathrm { f } ( 3 x )\). Given that $$\mathrm { f } ( x ) = 2 - \sqrt { x + 9 } , \quad x \in \mathbb { R } , \quad x \geq - 9 ,$$
2. find the values of \(a\) and \(b\),
3. find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.

5 The diagram shows the hyperbola $$\frac { x ^ { 2 } } { 4 } - y ^ { 2 } = 1$$ and its asymptotes. \includegraphics[max width=\textwidth, alt={}, center]{a0a30197-ca11-40d9-9ccd-30281c5e0fb4-03_531_1013_616_516}

1. Write down the equations of the two asymptotes.
2. The points on the hyperbola for which \(x = 4\) are denoted by \(A\) and \(B\). Find, in surd form, the \(y\)-coordinates of \(A\) and \(B\).
3. The hyperbola and its asymptotes are translated by two units in the positive \(y\) direction. Write down:
   1. the \(y\)-coordinates of the image points of \(A\) and \(B\) under this translation;
   2. the equations of the hyperbola and the asymptotes after the translation.

14 Curve \(C _ { 1 }\) has equation $$\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1$$ 14

1. Curve \(C _ { 2 }\) is a reflection of \(C _ { 1 }\) in the line \(y = x\) Write down an equation of \(C _ { 2 }\) 14
2. Curve \(C _ { 3 }\) is a circle of radius 4 , centred at the origin.
   Describe a single transformation which maps \(C _ { 1 }\) onto \(C _ { 3 }\) 14
3. Curve \(C _ { 4 }\) is a translation of \(C _ { 1 }\) The positive \(x\)-axis and the positive \(y\)-axis are tangents to \(C _ { 4 }\) 14 (c) (i) Sketch the graphs of \(C _ { 1 }\) and \(C _ { 4 }\) on the axes opposite. Indicate the coordinates of the \(x\) and \(y\) intercepts on your graphs.
   [0pt] [2 marks]
   14 (c) (ii) Determine the translation vector.
   [0pt] [2 marks]
   14 (c) (iii) The line \(y = m x + c\) is a tangent to both \(C _ { 1 }\) and \(C _ { 4 }\) Find the value of \(m\)

5

1. Show that the equation of \(H _ { 1 }\) can be written in the form $$( x - 1 ) ^ { 2 } - \frac { y ^ { 2 } } { q } = r$$ where \(q\) and \(r\) are integers.
   5
2. \(\quad \mathrm { H } _ { 2 }\) is the hyperbola $$x ^ { 2 } - y ^ { 2 } = 4$$ Describe fully a sequence of two transformations which maps the graph of \(H _ { 2 }\) onto the graph of \(H _ { 1 }\) [0pt] [4 marks]

6 The ellipse \(E _ { 1 }\) has equation $$x ^ { 2 } + \frac { y ^ { 2 } } { 4 } = 1$$ \(E _ { 1 }\) is translated by the vector \(\left[ \begin{array} { l } 3 \\ 0 \end{array} \right]\) to give the ellipse \(E _ { 2 }\) 6

1. Write down the equation of \(E _ { 2 }\) 6
2. The ellipse \(E _ { 3 }\) has equation $$\frac { x ^ { 2 } } { 4 } + ( y - 3 ) ^ { 2 } = 1$$ Describe the transformation that maps \(E _ { 2 }\) to \(E _ { 3 }\) 6
3. Each of the lines \(L _ { A }\) and \(L _ { B }\) is a tangent to both \(E _ { 2 }\) and \(E _ { 3 }\) \(L _ { A }\) is closer to the origin than \(L _ { B }\) \(E _ { 2 }\) and \(E _ { 3 }\) both lie between \(L _ { A }\) and \(L _ { B }\) Sketch and label \(E _ { 2 } , E _ { 3 } , L _ { A }\) and \(L _ { B }\) on the axes below.
   You do not need to show the values of the axis intercepts for \(L _ { A }\) and \(L _ { B }\) \includegraphics[max width=\textwidth, alt={}, center]{13abb93f-2fef-465c-980c-3b412de06618-09_1095_1095_726_475} 6
4. Explain, without doing any calculations, why \(L _ { A }\) has an equation of the form $$x + y = c$$ where \(c\) is a constant.

A curve \(C\) has parametric equations \(x = \sin \theta , y = \cos 2 \theta\). a) The equation of the tangent to the curve \(C\) at the point \(P\) where \(\theta = \frac { \pi } { 4 }\) is \(y = m x + c\). Find the exact values of \(m\) and \(c\).
b) Find the coordinates of the points of intersection of the curve \(C\) and the straight line \(x + y = 1\). The diagram below shows a sketch of the graph of \(y = f ( x )\). The graph crosses the \(y\)-axis at the point \(( 0 , - 2 )\), and the \(x\)-axis at the point \(( 8,0 )\). \includegraphics[max width=\textwidth, alt={}, center]{966abb82-ade0-4ca8-87a4-26e806d5add7-3_784_1080_1407_513}
a) Sketch the graph of \(y = - 4 f ( x + 3 )\). Indicate the coordinates of the point where the graph crosses the \(x\)-axis and the \(y\)-coordinate of the point where \(x = - 3\).
b) Sketch the graph of \(y = 3 + f ( 2 x )\). Indicate the \(y\)-coordinate of the point where \(x = 4\).

\includegraphics{figure_2} The diagram shows two curves. One curve has equation \(y = \sin x\) and the other curve has equation \(y = \text{f}(x)\).

1. In order to transform the curve \(y = \sin x\) to the curve \(y = \text{f}(x)\), the curve \(y = \sin x\) is first reflected in the \(x\)-axis. Describe fully a sequence of two further transformations which are required. [4]
2. Find f\((x)\) in terms of \(\sin x\). [2]

The curve \(y = x^2\) is transformed to the curve \(y = 4(x-3)^2 - 8\). Describe fully a sequence of transformations that have been combined, making clear the order in which the transformations have been applied. [5]

The equation of a curve is \(y = x^2 - 8x + 5\).

1. Find the coordinates of the minimum point of the curve. [2]

The curve is stretched by a factor of 2 parallel to the \(y\)-axis and then translated by \(\begin{pmatrix} 4 \\ 1 \end{pmatrix}\).

1. Find the coordinates of the minimum point of the transformed curve. [2]
2. Find the equation of the transformed curve. Give the answer in the form \(y = ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are integers to be found. [4]

\includegraphics{figure_5} In the diagram, the graph with equation \(y = \text{f}(x)\) is shown with solid lines and the graph with equation \(y = \text{g}(x)\) is shown with broken lines.

1. Describe fully a sequence of three transformations which transforms the graph of \(y = \text{f}(x)\) to the graph of \(y = \text{g}(x)\). [6]
2. Find an expression for g(x) in the form \(af(bx + c)\), where \(a\), \(b\) and \(c\) are integers. [2]

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = f(x), -1 \leq x \leq 3\). The curve touches the \(x\)-axis at the origin \(O\), crosses the \(x\)-axis at the point \(A(2, 0)\) and has a maximum at the point \(B(\frac{4}{3}, 1)\). In separate diagrams, show a sketch of the curve with equation

1. \(y = f(x + 1)\), [3]
2. \(y = |f(x)|\), [3]
3. \(y = f(|x|)\), [4]

marking on each sketch the coordinates of points at which the curve

1. has a turning point,
2. meets the \(x\)-axis.

The curve \(y = \ln x\) is transformed to the curve \(y = \ln(\frac{1}{2}x - a)\) by means of a translation followed by a stretch. It is given that \(a\) is a positive constant.

1. Give full details of the translation and stretch involved. [2]
2. Sketch the graph of \(y = \ln(\frac{1}{2}x - a)\). [2]
3. Sketch, on another diagram, the graph of \(y = |\ln(\frac{1}{2}x - a)|\). [2]
4. State, in terms of \(a\), the set of values of \(x\) for which \(|\ln(\frac{1}{2}x - a)| = -\ln(\frac{1}{2}x - a)\). [2]

The transformations R, S and T are defined as follows. \begin{align} \text{R} &: \text{ reflection in the } x\text{-axis}
\text{S} &: \text{ stretch in the } x\text{-direction with scale factor 3}
\text{T} &: \text{ translation in the positive } x\text{-direction by 4 units} \end{align}

1. The curve \(y = \ln x\) is transformed by R followed by T. Find the equation of the resulting curve. [2]
2. Find, in terms of S and T, a sequence of transformations that transforms the curve \(y = x^3\) to the curve \(y = \left(\frac{1}{3}x - 4\right)^3\). You should make clear the order of the transformations. [2]

The transformations \(\mathbf{R}\), \(\mathbf{S}\) and \(\mathbf{T}\) are defined as follows. \begin{align} \mathbf{R} &: \quad \text{reflection in the } x\text{-axis}
\mathbf{S} &: \quad \text{stretch in the } x\text{-direction with scale factor } 3
\mathbf{T} &: \quad \text{translation in the positive } x\text{-direction by } 4 \text{ units} \end{align}

1. The curve \(y = \ln x\) is transformed by \(\mathbf{R}\) followed by \(\mathbf{T}\). Find the equation of the resulting curve. [2]
2. Find, in terms of \(\mathbf{S}\) and \(\mathbf{T}\), a sequence of transformations that transforms the curve \(y = x^3\) to the curve \(y = \left(\frac{1}{3}x - 4\right)^3\). You should make clear the order of the transformations. [2]

A sequence of transformations maps the curve \(y = e^x\) to the curve \(y = e^{2x+3}\). Give details of these transformations. [3]

A cubic curve \(C\) has equation $$y = (3-x)(4+x)^2.$$

1. Sketch the graph of \(C\). [3] The sketch must include any points where the graph meets the coordinate axes.
2. Sketch in separate diagrams the graph of \(\ldots\)
   1. \(\ldots y = (3-2x)(4+2x)^2\). [2]
   2. \(\ldots y = (3+x)(4-x)^2\). [2]
   3. \(\ldots y = (2-x)(5+x)^2\). [2]
   Each of the sketches must include any points where the graph meets the coordinate axes.

A sequence of three transformations maps the curve \(y = \ln x\) to the curve \(y = \mathrm{e}^{3x} - 5\). Give details of these transformations. [4]