Finding quadratic constants from algebraic conditions

Determine unknown constants in a quadratic (or polynomial) expression from purely algebraic conditions such as a known minimum/stationary point, specific function values, or given roots, with no real-world context or graph of a trajectory involved.

8 questions · Moderate -0.6

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Edexcel P1 2023 January Q8
8 marks Moderate -0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb21001f-fe68-4776-992d-ede1aae233d7-20_728_885_248_584} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the straight line \(l\) and the curve \(C\).
Given that \(l\) cuts the \(y\)-axis at - 12 and cuts the \(x\)-axis at 4 , as shown in Figure 2,
  1. find an equation for \(l\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found. Given that \(C\)
    The region \(R\) is shown shaded in Figure 2.
  2. Use inequalities to define \(R\).
OCR MEI C1 Q11
12 marks Moderate -0.3
11 Fig. 11 shows the graph of \(y = a x ^ { 2 } + b x + c\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c556b8e-1a19-4480-bf2a-0ef9e67f98b4-4_572_1509_465_285} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. Explain why a must be negative.
  2. State two factors of \(y = a x ^ { 2 } + b x + c\).
  3. Hence, or otherwise, find the values of \(a , b\) and \(c\). Another function is given by \(y = x ^ { 2 } - 4 x + 10\).
  4. Write this in completed square form.
  5. Explain why the graphs of these two functions never meet.
OCR C1 Q2
4 marks Moderate -0.5
2. The curve \(C\) has the equation $$y = x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants.
Given that the minimum point of \(C\) has coordinates \(( - 2,5 )\), find the values of \(a\) and \(b\).
OCR MEI C1 2013 January Q5
4 marks Moderate -0.8
5 You are given that \(\mathrm { f } ( x ) = x ^ { 2 } + k x + c\).
Given also that \(\mathrm { f } ( 2 ) = 0\) and \(\mathrm { f } ( - 3 ) = 35\), find the values of the constants \(k\) and \(c\).
OCR PURE Q5
5 marks Moderate -0.8
5 A curve has equation \(y = a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants. The curve has a stationary point at \(( - 3,2 )\).
  1. State the values of \(b\) and \(c\). When the curve is translated by \(\binom { 4 } { 0 }\) the transformed curve passes through the point \(( 3 , - 18 )\).
  2. Determine the value of \(a\).
OCR MEI AS Paper 1 2024 June Q12
10 marks Moderate -0.8
12 The diagram shows the graph of \(\mathrm { f } ( \mathrm { x } ) = \mathrm { k } ( \mathrm { x } - \mathrm { p } ) ( \mathrm { x } - \mathrm { q } )\) where \(k , p\) and \(q\) are constants. The graph passes through the points \(( - 1,0 ) , ( 0 , - 4 )\) and \(( 2,0 )\). \includegraphics[max width=\textwidth, alt={}, center]{b5c47a93-ce43-4aa1-ba7f-fbb650523373-7_775_638_347_242}
  1. Find \(\mathrm { f } ( \mathrm { x } )\) in the form \(\mathrm { ax } ^ { 2 } + \mathrm { bx } + \mathrm { c }\). A cubic curve has gradient function \(f ( x )\). This cubic curve passes through the point \(( 0,8 )\).
  2. Find the equation of the cubic curve.
  3. Determine the coordinates of the stationary points of the cubic curve.
Edexcel C1 Q2
4 marks Moderate -0.5
2. The curve \(C\) has the equation $$y = x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants. Given that the minimum point of \(C\) has coordinates \(( - 2,5 )\), find the values of \(a\) and \(b\).
OCR MEI C1 Q8
4 marks Moderate -0.8
You are given that \(f(x) = x^2 + kx + c\). Given also that \(f(2) = 0\) and \(f(-3) = 35\), find the values of the constants \(k\) and \(c\). [4]