Polynomial with line intersection

Questions involving sketching a polynomial curve and a straight line, then finding their intersection points or analyzing their relationship.

8 questions · Moderate -0.8

1.02n Sketch curves: simple equations including polynomials1.03a Straight lines: equation forms y=mx+c, ax+by+c=0
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Edexcel C1 2012 January Q5
8 marks Moderate -0.8
5. The curve \(C\) has equation \(y = x ( 5 - x )\) and the line \(L\) has equation \(2 y = 5 x + 4\)
  1. Use algebra to show that \(C\) and \(L\) do not intersect.
  2. In the space on page 11, sketch \(C\) and \(L\) on the same diagram, showing the coordinates of the points at which \(C\) and \(L\) meet the axes.
Edexcel C1 2014 June Q4
5 marks Moderate -0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{64f015bf-29fb-4374-af34-3745ea49aced-05_945_1026_269_466} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$y = \frac { 1 } { x } + 1 , \quad x \neq 0$$ The curve \(C\) crosses the \(x\)-axis at the point \(A\).
  1. State the \(x\) coordinate of the point \(A\). The curve \(D\) has equation \(y = x ^ { 2 } ( x - 2 )\), for all real values of \(x\).
  2. A copy of Figure 1 is shown on page 7. On this copy, sketch a graph of curve \(D\).
    Show on the sketch the coordinates of each point where the curve \(D\) crosses the coordinate axes.
  3. Using your sketch, state, giving a reason, the number of real solutions to the equation $$x ^ { 2 } ( x - 2 ) = \frac { 1 } { x } + 1$$ \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{64f015bf-29fb-4374-af34-3745ea49aced-06_942_1026_516_466} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure}
Edexcel C1 Q6
14 marks Easy -1.2
6. $$f ( x ) = 9 - ( x - 2 ) ^ { 2 }$$
  1. Write down the maximum value of \(\mathrm { f } ( x )\).
  2. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of the points at which the graph meets the coordinate axes. The points \(A\) and \(B\) on the graph of \(y = \mathrm { f } ( x )\) have coordinates \(( - 2 , - 7 )\) and \(( 3,8 )\) respectively.
  3. Find, in the form \(y = m x + c\), an equation of the straight line through \(A\) and \(B\).
  4. Find the coordinates of the point at which the line \(A B\) crosses the \(x\)-axis. The mid-point of \(A B\) lies on the line with equation \(y = k x\), where \(k\) is a constant.
  5. Find the value of \(k\).
Edexcel C1 Q8
14 marks Easy -1.3
8. $$f ( x ) = 9 - ( x - 2 ) ^ { 2 }$$
  1. Write down the maximum value of \(\mathrm { f } ( x )\).
  2. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of the points at which the graph meets the coordinate axes. The points \(A\) and \(B\) on the graph of \(y = \mathrm { f } ( x )\) have coordinates \(( - 2 , - 7 )\) and \(( 3,8 )\) respectively.
  3. Find, in the form \(y = m x + c\), an equation of the straight line through \(A\) and \(B\).
  4. Find the coordinates of the point at which the line \(A B\) crosses the \(x\)-axis. The mid-point of \(A B\) lies on the line with equation \(y = k x\), where \(k\) is a constant.
  5. Find the value of \(k\).
AQA C1 2008 June Q1
11 marks Moderate -0.8
1 The straight line \(L\) has equation \(y = 3 x - 1\) and the curve \(C\) has equation $$y = ( x + 3 ) ( x - 1 )$$
  1. Sketch on the same axes the line \(L\) and the curve \(C\), showing the values of the intercepts on the \(x\)-axis and the \(y\)-axis.
  2. Show that the \(x\)-coordinates of the points of intersection of \(L\) and \(C\) satisfy the equation \(x ^ { 2 } - x - 2 = 0\).
  3. Hence find the coordinates of the points of intersection of \(L\) and \(C\).
Edexcel C1 Q42
9 marks Moderate -0.8
The curve \(C\) has equation \(y = x^2 - 4\) and the straight line \(l\) has equation \(y + 3x = 0\).
  1. In the space below, sketch \(C\) and \(l\) on the same axes. [3]
  2. Write down the coordinates of the points at which \(C\) meets the coordinate axes. [2]
  3. Using algebra, find the coordinates of the points at which \(l\) intersects \(C\). [4]
Edexcel C1 Q5
7 marks Moderate -0.8
  1. Sketch on the same diagram the graphs of \(y = (x - 1)^2(x - 5)\) and \(y = 8 - 2x\). Label on your diagram the coordinates of any points where each graph meets the coordinate axes. [5]
  2. Explain how your diagram shows that there is only one solution, \(\alpha\), to the equation $$(x - 1)^2(x - 5) = 8 - 2x.$$ [1]
  3. State the integer, \(n\), such that $$n < \alpha < n + 1.$$ [1]
OCR C1 Q4
7 marks Moderate -0.3
  1. Sketch on the same diagram the graphs of \(y = (x - 1)^2(x - 5)\) and \(y = 8 - 2x\). Label on your diagram the coordinates of any points where each graph meets the coordinate axes. [5]
  2. Explain how your diagram shows that there is only one solution, \(\alpha\), to the equation $$(x - 1)^2(x - 5) = 8 - 2x.$$ [1]
  3. State the integer, \(n\), such that $$n < \alpha < n + 1.$$ [1]