Polynomial with rational/modulus curves

Questions requiring sketching a polynomial alongside a rational function, modulus function, or piecewise function to analyze intersections.

3 questions

Edexcel P1 2022 October Q6
  1. (a) Given that \(k\) is a positive constant such that \(0 < k < 4\) sketch, on separate axes, the graphs of
    1. \(y = ( 2 x - k ) ( x + 4 ) ^ { 2 }\)
    2. \(y = \frac { k } { x ^ { 2 } }\)
      showing the coordinates of any points where the graphs cross or meet the coordinate axes, leaving coordinates in terms of \(k\), where appropriate.
      (b) State, with a reason, the number of roots of the equation
    $$( 2 x - k ) ( x + 4 ) ^ { 2 } = \frac { k } { x ^ { 2 } }$$
Edexcel P1 2023 October Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c0b4165d-b8bb-419c-b75a-d6c0c2431510-08_687_775_248_646} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation \(y = \frac { 1 } { x + 2 }\)
  1. State the equation of the asymptote of \(C\) that is parallel to the \(y\)-axis.
  2. Factorise fully \(x ^ { 3 } + 4 x ^ { 2 } + 4 x\) A copy of Figure 1, labelled Diagram 1, is shown on the next page.
  3. On Diagram 1, add a sketch of the curve with equation $$y = x ^ { 3 } + 4 x ^ { 2 } + 4 x$$ On your sketch, state clearly the coordinates of each point where this curve cuts or meets the coordinate axes.
  4. Hence state the number of real solutions of the equation $$( x + 2 ) \left( x ^ { 3 } + 4 x ^ { 2 } + 4 x \right) = 1$$ giving a reason for your answer.
    \includegraphics[max width=\textwidth, alt={}]{c0b4165d-b8bb-419c-b75a-d6c0c2431510-09_800_1700_1053_185}
    Only use the copy of Diagram 1 if you need to redraw your answer to part (c).
Edexcel AEA 2002 Specimen Q5
5．The function f is defined on the domain \([ - 2,2 ]\) by： $$f ( x ) = \left\{ \begin{array} { r l r } - k x ( 2 + x ) & \text { if } & - 2 \leq x < 0 ,
k x ( 2 - x ) & \text { if } & 0 \leq x \leq 2 , \end{array} \right.$$ where \(k\) is a positive constant．
The function g is defined on the domain \([ - 2,2 ]\) by \(\mathrm { g } ( x ) = ( 2.5 ) ^ { 2 } - x ^ { 2 }\) ．
（a）Prove that there is a value of \(k\) such that the graph of f touches the graph of g ．
（b）For this value of \(k\) sketch the graphs of the functions f and g on the same axes，stating clearly where the graphs touch．
（c）Find the exact area of the region bounded by the two graphs．