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 · Moderate -0.4

1.02w Graph transformations: simple transformations of f(x)
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OCR C1 Q8
9 marks Moderate -0.8
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
10 marks Standard +0.3
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
8 marks Moderate -0.8
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 H240/03 2023 June Q3
8 marks Moderate -0.3
The cubic polynomial \(\text{f}(x)\) is defined by \(\text{f}(x) = x^3 + px + q\), where \(p\) and \(q\) are constants.
    1. Given that \(\text{f}'(2) = 13\), find the value of \(p\). [2]
    2. Given also that \((x - 2)\) is a factor of \(\text{f}(x)\), find the value of \(q\). [2]
    The curve \(y = \text{f}(x)\) is translated by the vector \(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\).
  2. Using the values from part (a), determine the equation of the curve after it has been translated. Give your answer in the form \(y = x^3 + ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are integers to be found. [4]