Multiple transformations in sequence

Questions asking for the equation after applying two or more transformations in a specified order to a given curve.

4 questions

OCR C1 Q8
8. (i) Sketch the graphs of \(y = 2 x ^ { 4 }\) and \(y = 2 \sqrt { x } , x \geq 0\) on the same diagram and write down the coordinates of the point where they intersect.
(ii) Describe fully the transformation that maps the graph of \(y = 2 \sqrt { x }\) onto the graph of \(y = 2 \sqrt { x - 3 }\).
(iii) Find and simplify the equation of the graph obtained when the graph of \(y = 2 x ^ { 4 }\) is stretched by a factor of 2 in the \(x\)-direction, about the \(y\)-axis.
OCR C1 Q9
9. \(f ( x ) = 2 x ^ { 2 } + 3 x - 2\).
  1. Solve the equation \(\mathrm { f } ( x ) = 0\).
  2. Sketch the curve with equation \(y = \mathrm { f } ( x )\), showing the coordinates of any points of intersection with the coordinate axes.
  3. Find the coordinates of the points where the curve with equation \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\) crosses the coordinate axes. When the graph of \(y = \mathrm { f } ( x )\) is translated by 1 unit in the positive \(x\)-direction it maps onto the graph with equation \(y = a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are constants.
  4. Find the values of \(a , b\) and \(c\).
OCR MEI C3 Q4
4
  1. State the period of the function \(\mathrm { f } ( x ) = 1 + \cos 2 x\), where \(x\) is in degrees.
  2. State a sequence of two geometrical transformations which maps the curve \(y = \cos x\) onto the curve \(y = \mathrm { f } ( x )\).
  3. Sketch the graph of \(y = \mathrm { f } ( x )\) for \(- 180 ^ { \circ } < x < 180 ^ { \circ }\).
OCR C3 2010 June Q2
2 The transformations R, S and T are defined as follows.
R : reflection in the \(x\)-axis
S : stretch in the \(x\)-direction with scale factor 3
\(\mathrm { T } : \quad\) translation in the positive \(x\)-direction by 4 units
  1. The curve \(y = \ln x\) is transformed by R followed by T . Find the equation of the resulting curve.
  2. Find, in terms of S and T, a sequence of transformations that transforms the curve \(y = x ^ { 3 }\) to the curve \(y = \left( \frac { 1 } { 9 } x - 4 \right) ^ { 3 }\). You should make clear the order of the transformations.