1.02n Sketch curves: simple equations including polynomials

487 questions

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AQA C1 2006 January Q6
9 marks Moderate -0.8
6 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } + x ^ { 2 } - 10 x + 8$$
    1. Using the factor theorem, show that \(x - 2\) is a factor of \(\mathrm { p } ( x )\).
    2. Hence express \(\mathrm { p } ( x )\) as the product of three linear factors.
  2. Sketch the curve with equation \(y = x ^ { 3 } + x ^ { 2 } - 10 x + 8\), showing the coordinates of the points where the curve cuts the axes.
    (You are not required to calculate the coordinates of the stationary points.)
AQA C1 2009 January Q4
10 marks Easy -1.2
4
    1. Express \(x ^ { 2 } + 2 x + 5\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
    2. Hence show that \(x ^ { 2 } + 2 x + 5\) is always positive.
  2. A curve has equation \(y = x ^ { 2 } + 2 x + 5\).
    1. Write down the coordinates of the minimum point of the curve.
    2. Sketch the curve, showing the value of the intercept on the \(y\)-axis.
  3. Describe the geometrical transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 2 x + 5\).
AQA C1 2010 January Q5
11 marks Moderate -0.8
5
  1. Express \(( x - 5 ) ( x - 3 ) + 2\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
    (3 marks)
    1. Sketch the graph of \(y = ( x - 5 ) ( x - 3 ) + 2\), stating the coordinates of the minimum point and the point where the graph crosses the \(y\)-axis.
    2. Write down an equation of the tangent to the graph of \(y = ( x - 5 ) ( x - 3 ) + 2\) at its vertex.
  3. Describe the geometrical transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = ( x - 5 ) ( x - 3 ) + 2\).
AQA C1 2011 January Q5
13 marks Moderate -0.8
5
    1. Sketch the curve with equation \(y = x ( x - 2 ) ^ { 2 }\).
    2. Show that the equation \(x ( x - 2 ) ^ { 2 } = 3\) can be expressed as $$x ^ { 3 } - 4 x ^ { 2 } + 4 x - 3 = 0$$
  2. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 4 x - 3\).
    1. Find the remainder when \(\mathrm { p } ( x )\) is divided by \(x + 1\).
    2. Use the Factor Theorem to show that \(x - 3\) is a factor of \(\mathrm { p } ( x )\).
    3. Express \(\mathrm { p } ( x )\) in the form \(( x - 3 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
  3. Hence show that the equation \(x ( x - 2 ) ^ { 2 } = 3\) has only one real root and state the value of this root.
AQA C1 2012 January Q2
10 marks Moderate -0.8
2
  1. Factorise \(x ^ { 2 } - 4 x - 12\).
  2. Sketch the graph with equation \(y = x ^ { 2 } - 4 x - 12\), stating the values where the curve crosses the coordinate axes.
    1. Express \(x ^ { 2 } - 4 x - 12\) in the form \(( x - p ) ^ { 2 } - q\), where \(p\) and \(q\) are positive integers.
    2. Hence find the minimum value of \(x ^ { 2 } - 4 x - 12\).
  4. The curve with equation \(y = x ^ { 2 } - 4 x - 12\) is translated by the vector \(\left[ \begin{array} { r } - 3 \\ 2 \end{array} \right]\). Find an equation of the new curve. You need not simplify your answer.
AQA C1 2013 January Q4
12 marks Moderate -0.8
4
    1. Express \(x ^ { 2 } - 6 x + 11\) in the form \(( x - p ) ^ { 2 } + q\).
    2. Use the result from part (a)(i) to show that the equation \(x ^ { 2 } - 6 x + 11 = 0\) has no real solutions.
  2. A curve has equation \(y = x ^ { 2 } - 6 x + 11\).
    1. Find the coordinates of the vertex of the curve.
    2. Sketch the curve, indicating the value of \(y\) where the curve crosses the \(y\)-axis.
    3. Describe the geometrical transformation that maps the curve with equation \(y = x ^ { 2 } - 6 x + 11\) onto the curve with equation \(y = x ^ { 2 }\).
AQA C1 2005 June Q2
10 marks Easy -1.2
2
  1. Express \(x ^ { 2 } - 6 x + 16\) in the form \(( x - p ) ^ { 2 } + q\).
  2. A curve has equation \(y = x ^ { 2 } - 6 x + 16\). Using your answer from part (a), or otherwise:
    1. find the coordinates of the vertex (minimum point) of the curve;
    2. sketch the curve, indicating the value where the curve crosses the \(y\)-axis;
    3. state the equation of the line of symmetry of the curve.
  3. Describe geometrically the transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } - 6 x + 16\).
AQA C1 2006 June Q2
10 marks Moderate -0.8
2
  1. Express \(x ^ { 2 } + 8 x + 19\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
  2. Hence, or otherwise, show that the equation \(x ^ { 2 } + 8 x + 19 = 0\) has no real solutions.
  3. Sketch the graph of \(y = x ^ { 2 } + 8 x + 19\), stating the coordinates of the minimum point and the point where the graph crosses the \(y\)-axis.
  4. Describe geometrically the transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 8 x + 19\).
AQA C1 2010 June Q3
12 marks Moderate -0.8
3 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } + 7 x ^ { 2 } + 7 x - 15$$
    1. Use the Factor Theorem to show that \(x + 3\) is a factor of \(\mathrm { p } ( x )\).
    2. Express \(\mathrm { p } ( x )\) as the product of three linear factors.
  2. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 2\).
    1. Verify that \(\mathrm { p } ( - 1 ) < \mathrm { p } ( 0 )\).
    2. Sketch the curve with equation \(y = x ^ { 3 } + 7 x ^ { 2 } + 7 x - 15\), indicating the values where the curve crosses the coordinate axes.
AQA C1 2011 June Q4
12 marks Easy -1.2
4
  1. Express \(x ^ { 2 } + 5 x + 7\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
    (3 marks)
  2. A curve has equation \(y = x ^ { 2 } + 5 x + 7\).
    1. Find the coordinates of the vertex of the curve.
    2. State the equation of the line of symmetry of the curve.
    3. Sketch the curve, stating the value of the intercept on the \(y\)-axis.
  3. Describe the geometrical transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 5 x + 7\).
AQA C1 2012 June Q3
10 marks Moderate -0.8
3 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 5 x - 6$$
    1. Use the Factor Theorem to show that \(x + 1\) is a factor of \(\mathrm { p } ( x )\).
    2. Express \(\mathrm { p } ( x )\) as the product of three linear factors.
  2. Verify that \(\mathrm { p } ( 0 ) > \mathrm { p } ( 1 )\).
  3. Sketch the curve with equation \(y = x ^ { 3 } + 2 x ^ { 2 } - 5 x - 6\), indicating the values where the curve crosses the \(x\)-axis.
AQA C1 2014 June Q4
7 marks Moderate -0.8
4
    1. Express \(16 - 6 x - x ^ { 2 }\) in the form \(p - ( x + q ) ^ { 2 }\) where \(p\) and \(q\) are integers.
    2. Hence write down the maximum value of \(16 - 6 x - x ^ { 2 }\).
    1. Factorise \(16 - 6 x - x ^ { 2 }\).
    2. Sketch the curve with equation \(y = 16 - 6 x - x ^ { 2 }\), stating the values of \(x\) where the curve crosses the \(x\)-axis and the value of the \(y\)-intercept.
      [0pt] [3 marks]
AQA C1 2015 June Q7
12 marks Moderate -0.8
7
  1. Sketch the curve with equation \(y = x ^ { 2 } ( x - 3 )\).
  2. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 2 } ( x - 3 ) + 20\).
    1. Find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 4\).
    2. Use the Factor Theorem to show that \(x + 2\) is a factor of \(\mathrm { p } ( x )\).
    3. Express \(\mathrm { p } ( x )\) in the form \(( x + 2 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
    4. Hence show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root and state its value.
      [0pt] [3 marks]
AQA C1 2016 June Q6
8 marks Standard +0.3
6
  1. A curve has equation \(y = 8 - 4 x - 2 x ^ { 2 }\).
    1. Find the values of \(x\) where the curve crosses the \(x\)-axis, giving your answer in the form \(m \pm \sqrt { n }\), where \(m\) and \(n\) are integers.
    2. Sketch the curve, giving the value of the \(y\)-intercept.
  2. A line has equation \(y = k ( x + 4 )\), where \(k\) is a constant.
    1. Show that the \(x\)-coordinates of any points of intersection of the line with the curve \(y = 8 - 4 x - 2 x ^ { 2 }\) satisfy the equation $$2 x ^ { 2 } + ( k + 4 ) x + 4 ( k - 2 ) = 0$$
    2. Find the values of \(k\) for which the line is a tangent to the curve \(y = 8 - 4 x - 2 x ^ { 2 }\).
      [0pt] [3 marks]
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 Q6
5 marks Moderate -0.8
6. Given that \(\mathrm { f } ( x ) = 15 - 7 x - 2 x ^ { 2 }\),
  1. find the coordinates of all points at which the graph of \(y = \mathrm { f } ( x )\) crosses the coordinate axes.
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
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 \(k\).
Edexcel C1 Q5
6 marks Moderate -0.8
5. $$f ( x ) = 4 x - 3 x ^ { 2 } - x ^ { 3 }$$
  1. Fully factorise \(4 x - 3 x ^ { 2 } - x ^ { 3 }\).
  2. Sketch the curve \(y = \mathrm { f } ( x )\), showing the coordinates of any points of intersection with the coordinate axes.
Edexcel C1 Q10
14 marks Moderate -0.3
10. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = ( x + 2 ) ^ { 3 }$$
  1. Sketch the curve \(C\), showing the coordinates of any points of intersection with the coordinate axes.
  2. Find f \({ } ^ { \prime } ( x )\). The straight line \(l\) is the tangent to \(C\) at the point \(P ( - 1,1 )\).
  3. Find an equation for \(l\). The straight line \(m\) is parallel to \(l\) and is also a tangent to \(C\).
  4. Show that \(m\) has the equation \(y = 3 x + 8\).
Edexcel C1 Q5
10 marks Moderate -0.3
5. The curve \(C\) with equation \(y = ( 2 - x ) ( 3 - x ) ^ { 2 }\) crosses the \(x\)-axis at the point \(A\) and touches the \(x\)-axis at the point \(B\).
  1. Sketch the curve \(C\), showing the coordinates of \(A\) and \(B\).
  2. Show that the tangent to \(C\) at \(A\) has the equation $$x + y = 2 .$$
Edexcel C1 Q5
7 marks Moderate -0.8
  1. The curve \(y = \mathrm { f } ( x )\) passes through the point \(P ( - 1,3 )\) and is such that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 1 } { x ^ { 2 } } , \quad x \neq 0 .$$
  1. Using integration, find \(\mathrm { f } ( x )\).
  2. Sketch the curve \(y = \mathrm { f } ( x )\) and write down the equations of its asymptotes.
Edexcel C2 Q3
8 marks Moderate -0.8
3. Given that \(\mathrm { f } ( x ) = 15 - 7 x - 2 x ^ { 2 }\),
  1. find the coordinates of all points at which the graph of \(y = \mathrm { f } ( x )\) crosses the coordinate axes.
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
  3. Calculate the coordinates of the stationary point of \(\mathrm { f } ( x )\).
    [0pt] [P1 June 2002 Question 3]
Edexcel C2 Q9
13 marks Standard +0.3
9. $$f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 16 .$$
  1. Evaluate \(\mathrm { f } ( 1 )\) and hence state a linear factor of \(\mathrm { f } ( x )\).
  2. Show that \(\mathrm { f } ( x )\) can be expressed in the form $$\mathrm { f } ( x ) = ( x + p ) ( x + q ) ^ { 2 } ,$$ where \(p\) and \(q\) are integers to be found.
  3. Sketch the curve \(y = \mathrm { f } ( x )\).
  4. Using integration, find the area of the region enclosed by the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis.
Edexcel C2 Q7
11 marks Moderate -0.3
7. $$f ( x ) = 2 + 6 x ^ { 2 } - x ^ { 3 }$$
  1. Find the coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\).
  2. Determine whether each stationary point is a maximum or minimum point.
  3. Sketch the curve \(y = \mathrm { f } ( x )\).
  4. State the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = k\) has three solutions.
AQA C3 2005 June Q6
13 marks Moderate -0.3
6
    1. Sketch the graph of \(y = 4 - x ^ { 2 }\), indicating the coordinates of the points where the graph crosses the coordinate axes.
    2. The region between the graph and the \(x\)-axis from \(x = 0\) to \(x = 2\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the exact value of the volume of the solid generated.
    1. Sketch the graph of \(y = \left| 4 - x ^ { 2 } \right|\).
    2. Solve \(\left| 4 - x ^ { 2 } \right| = 3\).
    3. Hence, or otherwise, solve the inequality \(\left| 4 - x ^ { 2 } \right| < 3\).