Polynomial with line intersection

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

7 questions

Edexcel C1 2012 January Q5
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
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
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
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 \(k\).
Edexcel C1 Q6
6. (a) Sketch on the same diagram the curve with equation \(y = ( x - 2 ) ^ { 2 }\) and the straight line with equation \(y = 2 x - 1\). Label on your sketch the coordinates of any points where each graph meets the coordinate axes.
(b) Find the set of values of \(x\) for which $$( x - 2 ) ^ { 2 } > 2 x - 1$$
Edexcel C1 Q19
19. $$f ( x ) = 9 - ( x - 2 ) ^ { 2 }$$
  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
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 \(C\).