Identify/describe sequence of transformations between two given equations

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1. The curve with equation \(y = x ^ { 2 } + 2 x - 5\) is translated by \(\binom { - 1 } { 3 }\).
   Find the equation of the translated curve, giving your answer in the form \(y = a x ^ { 2 } + b x + c\).
2. The curve with equation \(y = x ^ { 2 } + 2 x - 5\) is transformed to a curve with equation \(y = 4 x ^ { 2 } + 4 x - 5\). Describe fully the single transformation that has been applied.

6 \includegraphics[max width=\textwidth, alt={}, center]{32f90420-e1eb-47ab-b588-e3806b64813f-3_641_837_1306_657} The diagram shows the graph of \(y = - \sin ^ { - 1 } ( x - 1 )\).

1. Give details of the pair of geometrical transformations which transforms the graph of \(y = - \sin ^ { - 1 } ( x - 1 )\) to the graph of \(y = \sin ^ { - 1 } x\).
2. Sketch the graph of \(y = \left| - \sin ^ { - 1 } ( x - 1 ) \right|\).
3. Find the exact solutions of the equation \(\left| - \sin ^ { - 1 } ( x - 1 ) \right| = \frac { 1 } { 3 } \pi\).

7 \includegraphics[max width=\textwidth, alt={}, center]{c940af95-e291-402a-856c-9090d13163d5-3_419_700_1809_721} The diagram shows the curve \(y = \mathrm { e } ^ { k x } - a\), where \(k\) and \(a\) are constants.

1. Give details of the pair of transformations which transforms the curve \(y = \mathrm { e } ^ { x }\) to the curve \(y = \mathrm { e } ^ { k x } - a\).
2. Sketch the curve \(y = \left| \mathrm { e } ^ { k x } - a \right|\).
3. Given that the curve \(y = \left| \mathrm { e } ^ { k x } - a \right|\) passes through the points \(( 0,13 )\) and \(( \ln 3,13 )\), find the values of \(k\) and \(a\).

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]

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

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

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

Describe a sequence of transformations which maps the graph of $$y = |2x - 5|$$ onto the graph of $$y = |x|$$ [3 marks]

The graph of \(y = \text{e}^x\) can be transformed to the graph of \(y = \text{e}^{2x-1}\) by a stretch parallel to the \(x\)-axis followed by a translation.

1. 1. State the scale factor of the stretch. [1]
   2. Give full details of the translation. [2]

Alternatively the graph of \(y = \text{e}^x\) can be transformed to the graph of \(y = \text{e}^{2x-1}\) by a stretch parallel to the \(x\)-axis and a stretch parallel to the \(y\)-axis.

1. State the scale factor of the stretch parallel to the \(y\)-axis. [1]

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]