Transformation effect on key points

Questions that ask for the coordinates of a transformed point (typically a minimum or maximum) after applying a given transformation to a curve.

4 questions · Moderate -0.5

1.02w Graph transformations: simple transformations of f(x)1.07i Differentiate x^n: for rational n and sums
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Edexcel C12 2014 June Q4
8 marks Moderate -0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b85872d4-00b2-499b-9765-f7559d3de66a-05_716_725_219_603} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = x ^ { 2 } + \frac { 16 } { x } , \quad x > 0$$ The curve has a minimum turning point at \(A\).
  1. Find \(\mathrm { f } ^ { \prime } ( x )\).
  2. Hence find the coordinates of \(A\).
  3. Use your answer to part (b) to write down the turning point of the curve with equation
    1. \(y = \mathrm { f } ( x + 1 )\),
    2. \(y = \frac { 1 } { 2 } \mathrm { f } ( x )\).
OCR MEI C2 2011 January Q4
4 marks Easy -1.2
4 The curve \(y = \mathrm { f } ( x )\) has a minimum point at \(( 3,5 )\).
State the coordinates of the corresponding minimum point on the graph of
  1. \(y = 3 \mathrm { f } ( x )\),
  2. \(y = \mathrm { f } ( 2 x )\).
OCR MEI C2 Q3
12 marks Standard +0.3
  1. Express \(\mathrm { f } ( x )\) in factorised form.
  2. Show that the equation of the curve may be written as \(y = x ^ { 3 } + 5 x ^ { 2 } - 4 x - 20\).
  3. Use calculus to show that, correct to 1 decimal place, the \(x\)-coordinate of the minimum point on the curve is 0.4 . Find also the coordinates of the maximum point on the curve, giving your answers correct to 1 decimal place.
  4. State, correct to 1 decimal place, the coordinates of the maximum point on the curve \(y = \mathrm { f } ( 2 x )\).
Edexcel PURE 2024 October Q9
Moderate -0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c48e6503-9d26-4f55-bdca-feadfb1afb7c-26_732_730_251_669} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( x + 5 ) \left( 3 x ^ { 2 } - 4 x + 20 \right)$$
  1. Deduce the range of values of \(x\) for which \(\mathrm { f } ( x ) \geqslant 0\)
  2. Find \(\mathrm { f } ^ { \prime } ( x )\) giving your answer in simplest form. The point \(R ( - 4,84 )\) lies on \(C\).
    Given that the tangent to \(C\) at the point \(P\) is parallel to the tangent to \(C\) at the point \(R\) (c) find the \(x\) coordinate of \(P\).
    (d) Find the point to which \(R\) is transformed when the curve with equation \(y = \mathrm { f } ( x )\) is transformed to the curve with equation,
    1. \(y = \mathrm { f } ( x - 3 )\)
    2. \(y = 4 \mathrm { f } ( x )\)