Sketch transformations from algebraic function

Questions that give an explicit algebraic function (like f(x) = x³ - 6x² + 5x + 12 or f(x) = ln x) and ask students to sketch the original and/or transformed versions, requiring both algebraic manipulation and transformation application.

5 questions

Edexcel C3 2013 June Q2
2. Given that $$\mathrm { f } ( x ) = \ln x , \quad x > 0$$ sketch on separate axes the graphs of
  1. \(\quad y = \mathrm { f } ( x )\),
  2. \(y = | \mathrm { f } ( x ) |\),
  3. \(y = - \mathrm { f } ( x - 4 )\). Show, on each diagram, the point where the graph meets or crosses the \(x\)-axis. In each case, state the equation of the asymptote.
OCR C1 Q8
8. $$f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12$$
  1. Show that $$( x + 1 ) ( x - 3 ) ( x - 4 ) \equiv x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12 .$$
  2. Sketch the curve \(y = \mathrm { f } ( x )\), showing the coordinates of any points of intersection with the coordinate axes.
  3. Showing the coordinates of any points of intersection with the coordinate axes, sketch on separate diagrams the curves
    1. \(\quad y = \mathrm { f } ( x + 3 )\),
    2. \(y = \mathrm { f } ( - x )\).
OCR C3 2009 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{6a690aa5-63a7-4569-afa8-0746814ebab4-4_648_1132_262_504} The diagram shows the curves \(y = \ln x\) and \(y = 2 \ln ( x - 6 )\). The curves meet at the point \(P\) which has \(x\)-coordinate \(a\). The shaded region is bounded by the curve \(y = 2 \ln ( x - 6 )\) and the lines \(x = a\) and \(y = 0\).
  1. Give details of the pair of transformations which transforms the curve \(y = \ln x\) to the curve \(y = 2 \ln ( x - 6 )\).
  2. Solve an equation to find the value of \(a\).
  3. Use Simpson's rule with two strips to find an approximation to the area of the shaded region.
OCR MEI C3 2009 January Q5
5
  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 \(- 180 ^ { \circ } < x < 180 ^ { \circ }\).
Edexcel C1 Q8
8. $$f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12$$
  1. Show that $$( x + 1 ) ( x - 3 ) ( x - 4 ) \equiv x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12$$
  2. Sketch the curve \(y = \mathrm { f } ( x )\), showing the coordinates of any points of intersection with the coordinate axes.
  3. Showing the coordinates of any points of intersection with the coordinate axes, sketch on separate diagrams the curves
    1. \(\quad y = \mathrm { f } ( x + 3 )\),
    2. \(y = \mathrm { f } ( - x )\).