Find constants from sketch features

Questions showing a sketch with marked features (intercepts, turning points) where students must determine unknown constants in a partially given polynomial equation.

3 questions · Standard +0.6

1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02w Graph transformations: simple transformations of f(x)
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Edexcel C1 2013 June Q9
8 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cfc23548-bf4f-4efa-9ceb-b8d03bb1f019-13_698_1413_118_280} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\).
The curve \(C\) passes through the point \(( - 1,0 )\) and touches the \(x\)-axis at the point \(( 2,0 )\).
The curve \(C\) has a maximum at the point ( 0,4 ).
  1. The equation of the curve \(C\) can be written in the form $$y = x ^ { 3 } + a x ^ { 2 } + b x + c$$ where \(a\), \(b\) and \(c\) are integers.
    Calculate the values of \(a , b\) and \(c\).
  2. Sketch the curve with equation \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\) in the space provided on page 24 Show clearly the coordinates of all the points where the curve crosses or meets the coordinate axes.
Edexcel P3 2021 January Q2
6 marks Moderate -0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{624e9e2f-b6b8-47ce-accc-31dcd5f0554e-04_903_1148_123_399} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where \(x \in \mathbb { R }\) and \(\mathrm { f } ( x )\) is a polynomial. The curve passes through the origin and touches the \(x\)-axis at the point \(( 3,0 )\) There is a maximum turning point at \(( 1,2 )\) and a minimum turning point at \(( 3,0 )\) On separate diagrams, sketch the curve with equation
  1. \(y = 3 f ( 2 x )\)
  2. \(y = \mathrm { f } ( - x ) - 1\) On each sketch, show clearly the coordinates of
    • the point where the curve crosses the \(y\)-axis
    • any maximum or minimum turning points
Edexcel AEA 2012 June Q6
16 marks Challenging +1.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fc5d0d07-b750-4646-bdcb-419a290200c9-4_433_1011_221_529} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = ( x + a ) ( x - b ) ^ { 2 }\), where \(a\) and \(b\) are positive constants. The curve cuts the \(x\)-axis at \(P\) and has a maximum point at \(S\) and a minimum point at \(Q\).
  1. Write down the coordinates of \(P\) and \(Q\) in terms of \(a\) and \(b\).
  2. Show that \(G\), the area of the shaded region between the curve \(P S Q\) and the \(x\)-axis, is given by \(G = \frac { ( a + b ) ^ { 4 } } { 12 }\). The rectangle \(P Q R S T\) has \(R S T\) parallel to \(Q P\) and both \(P T\) and \(Q R\) are parallel to the \(y\)-axis.
  3. Show that \(\frac { G } { \text { Area of } P Q R S T } = k\), where \(k\) is a constant independent of \(a\) and \(b\) and find the value of \(k\).