Graph to graph transformation description

3
\includegraphics[max width=\textwidth, alt={}, center]{77f27b11-b931-481f-b4ef-5e549eff8086-04_1150_1164_269_484} The diagram shows graphs with equations \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\).
Describe fully a sequence of two transformations which transforms the graph of \(y = \mathrm { f } ( x )\) to \(y = \mathrm { g } ( x )\).

1
\includegraphics[max width=\textwidth, alt={}, center]{51bd3ba6-e1d1-4c07-81cd-d99dd77f9306-02_778_1061_269_532} The diagram shows the graph of \(y = \mathrm { f } ( x )\), which consists of the two straight lines \(A B\) and \(B C\). The lines \(A ^ { \prime } B ^ { \prime }\) and \(B ^ { \prime } C ^ { \prime }\) form the graph of \(y = \mathrm { g } ( x )\), which is the result of applying a sequence of two transformations, in either order, to \(y = \mathrm { f } ( x )\). State fully the two transformations.

5
\includegraphics[max width=\textwidth, alt={}, center]{54f3f051-e124-470d-87b5-8e25c35248a9-07_775_768_260_685} In the diagram, the graph of \(y = \mathrm { f } ( x )\) is shown with solid lines. The graph shown with broken lines is a transformation of \(y = \mathrm { f } ( x )\).

1. Describe fully the two single transformations of \(y = \mathrm { f } ( x )\) that have been combined to give the resulting transformation.
2. State in terms of \(y\), f and \(x\), the equation of the graph shown with broken lines.

5
\includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-06_631_1500_260_285} In the diagram, the graph with equation \(y = \mathrm { f } ( x )\) is shown with solid lines and the graph with equation \(y = \mathrm { g } ( x )\) is shown with broken lines.

1. Describe fully a sequence of three transformations which transforms the graph of \(y = \mathrm { f } ( x )\) to the graph of \(y = \mathrm { g } ( x )\).
2. Find an expression for \(\mathrm { g } ( x )\) in the form \(a \mathrm { f } ( b x + c )\), where \(a , b\) and \(c\) are integers.
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   \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-06_2716_31_106_2016}
   \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-07_2723_35_101_20}

3 Each of the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\) 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 \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\).

1. 
   \includegraphics[max width=\textwidth, alt={}, center]{8350e810-3ceb-4876-a7a8-249e17616057-2_433_716_569_710}
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{8350e810-3ceb-4876-a7a8-249e17616057-2_396_612_1130_761}