Describe transformation from graph

3 In each of parts (a), (b) and (c), the graph shown with solid lines has equation \(y = \mathrm { f } ( x )\). The graph shown with broken lines is a transformation of \(y = \mathrm { f } ( x )\).

1. \includegraphics[max width=\textwidth, alt={}, center]{aa4c496d-ce5f-4f46-ad37-d901644a9e7c-04_412_645_367_788} State, in terms of f , the equation of the graph shown with broken lines.
2. \includegraphics[max width=\textwidth, alt={}, center]{aa4c496d-ce5f-4f46-ad37-d901644a9e7c-04_650_423_1046_900} State, in terms of f , the equation of the graph shown with broken lines.
3. \includegraphics[max width=\textwidth, alt={}, center]{aa4c496d-ce5f-4f46-ad37-d901644a9e7c-04_550_631_1975_804} State, in terms of f , the equation of the graph shown with broken lines.

2 \begin{figure}[h] \end{figure} Fig. 2 shows graphs \(A\) and \(B\).

1. State the transformation which maps graph \(A\) onto graph \(B\).
2. The equation of graph \(A\) is \(y = \mathrm { f } ( x )\). Which one of the following is the equation of graph \(B\) ? $$\begin{aligned} & y = \mathrm { f } ( x ) + 2 \\ & y = 2 \mathrm { f } ( x ) \end{aligned}$$ $$\begin{aligned} & y = \mathrm { f } ( x ) - 2 \\ & y = \mathrm { f } ( x + 3 ) \end{aligned}$$ $$\begin{aligned} & y = \mathrm { f } ( x + 2 ) \\ & y = \mathrm { f } ( x - 3 ) \end{aligned}$$

13 Prove by induction that, for all integers \(n \geq 1\) $$\sum _ { r = 1 } ^ { n } 2 ^ { - r } = 1 - 2 ^ { - n }$$ [4 marks]

\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]

\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_2} Fig. 2 shows graphs \(A\) and \(B\).

1. State the transformation which maps graph \(A\) onto graph \(B\). [2]
2. The equation of graph \(A\) is \(y = f(x)\). Which one of the following is the equation of graph \(B\)? \(y = f(x) + 2\) \quad \(y = f(x) - 2\) \quad \(y = f(x + 2)\) \quad \(y = f(x - 2)\) \(y = 2f(x)\) \quad \(y = f(x + 3)\) \quad \(y = f(x - 3)\) \quad \(y = 3f(x)\) [2]

Each of the graphs of \(y = \text{f}(x)\) and \(y = \text{g}(x)\) below is obtained using a sequence of two transformations applied to the corresponding dashed graph. In each case, state suitable transformations, and hence find expressions for \(\text{f}(x)\) and \(\text{g}(x)\).

1. \includegraphics{figure_5i} [3]
2. \includegraphics{figure_5ii} [3]

Each of the graphs of \(y = \text{f}(x)\) and \(y = \text{g}(x)\) below is obtained using a sequence of two transformations applied to the corresponding dashed graph. In each case, state suitable transformations, and hence find expressions for f(x) and g(x).

1. \includegraphics{figure_3i} [3]
2. \includegraphics{figure_3ii} [3]