1.02n Sketch curves: simple equations including polynomials

487 questions

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Edexcel C1 Q9
8 marks Moderate -0.8
Given that \(f(x) = (x^2 - 6x)(x - 2) + 3x\),
  1. express \(f(x)\) in the form \(a(x^2 + bx + c)\), where \(a\), \(b\) and \(c\) are constants. [3]
  2. Hence factorise \(f(x)\) completely. [2]
  3. Sketch the graph of \(y = f(x)\), showing the coordinates of each point at which the graph meets the axes. [3]
Edexcel C1 Q10
13 marks Moderate -0.3
  1. On the same axes sketch the graphs of the curves with equations
    1. \(y = x^2(x - 2)\), [3]
    2. \(y = x(6 - x)\), [3]
    and indicate on your sketches the coordinates of all the points where the curves cross the \(x\)-axis.
  2. Use algebra to find the coordinates of the points where the graphs intersect. [7]
Edexcel C1 Q14
5 marks Moderate -0.8
Given that \(f(x) = 15 - 7x - 2x^2\),
  1. find the coordinates of all points at which the graph of \(y = f(x)\) crosses the coordinate axes. [3]
  2. Sketch the graph of \(y = f(x)\). [2]
Edexcel C1 Q19
14 marks Easy -1.2
\(f(x) = 9 - (x - 2)^2\)
  1. Write down the maximum value of \(f(x)\). [1]
  2. Sketch the graph of \(y = f(x)\), showing the coordinates of the points at which the graph meets the coordinate axes. [5]
The points \(A\) and \(B\) on the graph of \(y = f(x)\) have coordinates \((-2, -7)\) and \((3, 8)\) respectively.
  1. Find, in the form \(y = mx + c\), an equation of the straight line through \(A\) and \(B\). [4]
  2. Find the coordinates of the point at which the line \(AB\) crosses the \(x\)-axis. [2]
The mid-point of \(AB\) lies on the line with equation \(y = kx\), where \(k\) is a constant.
  1. Find the value of \(k\). [2]
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 Specimen Q8
11 marks Easy -1.2
Given that $$x^2 + 10x + 36 = (x + a)^2 + b$$ where \(a\) and \(b\) are constants,
  1. find the value of \(a\) and the value of \(b\). [3]
  2. Hence show that the equation \(x^2 + 10x + 36 = 0\) has no real roots. [2]
The equation \(x^2 + 10x + k = 0\) has equal roots.
  1. Find the value of \(k\). [2]
  2. For this value of \(k\), sketch the graph of \(y = x^2 + 10x + k\), showing the coordinates of any points at which the graph meets the coordinate axes. [4]
Edexcel C2 Q23
8 marks Moderate -0.8
Given that \(f(x) = 15 - 7x - 2x^2\),
  1. find the coordinates of all points at which the graph of \(y = f(x)\) crosses the coordinate axes. [3]
  2. Sketch the graph of \(y = f(x)\). [2]
  3. Calculate the coordinates of the stationary point of \(f(x)\). [3]
OCR C1 2013 January Q3
5 marks Moderate -0.3
  1. Sketch the curve \(y = (1 + x)(2 - x)(3 + x)\), giving the coordinates of all points of intersection with the axes. [3]
  2. Describe the transformation that transforms the curve \(y = (1 + x)(2 - x)(3 + x)\) to the curve \(y = (1 - x)(2 + x)(3 - x)\). [2]
OCR C1 2006 June Q4
8 marks Easy -1.2
  1. By expanding the brackets, show that $$(x - 4)(x - 3)(x + 1) = x^3 - 6x^2 + 5x + 12.$$ [3]
  2. Sketch the curve $$y = x^3 - 6x^2 + 5x + 12,$$ giving the coordinates of the points where the curve meets the axes. Label the curve \(C_1\). [3]
  3. On the same diagram as in part (ii), sketch the curve $$y = -x^3 + 6x^2 - 5x - 12.$$ Label this curve \(C_2\). [2]
OCR C1 2013 June Q9
12 marks Moderate -0.8
  1. Sketch the curve \(y = 2x^2 - x - 6\), giving the coordinates of all points of intersection with the axes. [5]
  2. Find the set of values of \(x\) for which \(2x^2 - x - 6\) is a decreasing function. [3]
  3. The line \(y = 4\) meets the curve \(y = 2x^2 - x - 6\) at the points \(P\) and \(Q\). Calculate the distance \(PQ\). [4]
OCR C1 2014 June Q10
12 marks Moderate -0.3
A curve has equation \(y = (x + 2)^2(2x - 3)\).
  1. Sketch the curve, giving the coordinates of all points of intersection with the axes. [3]
  2. Find an equation of the tangent to the curve at the point where \(x = -1\). Give your answer in the form \(ax + by + c = 0\). [9]
OCR MEI C1 Q7
5 marks Easy -1.2
Express \(x^2 - 6x\) in the form \((x - a)^2 - b\). Sketch the graph of \(y = x^2 - 6x\), giving the coordinates of its minimum point and the intersections with the axes. [5]
OCR MEI C1 Q11
12 marks Moderate -0.3
A cubic polynomial is given by \(f(x) = x^3 + x^2 - 10x + 8\).
  1. Show that \((x - 1)\) is a factor of \(f(x)\). Factorise \(f(x)\) fully. Sketch the graph of \(y = f(x)\). [7]
  2. The graph of \(y = f(x)\) is translated by \(\begin{pmatrix} -3 \\ 0 \end{pmatrix}\). Write down an equation for the resulting graph. You need not simplify your answer. Find also the intercept on the \(y\)-axis of the resulting graph. [5]
OCR MEI C1 2006 January Q11
13 marks Moderate -0.8
  1. Write \(x^2 - 7x + 6\) in the form \((x - a)^2 + b\). [3]
  2. State the coordinates of the minimum point on the graph of \(y = x^2 - 7x + 6\). [2]
  3. Find the coordinates of the points where the graph of \(y = x^2 - 7x + 6\) crosses the axes and sketch the graph. [5]
  4. Show that the graphs of \(y = x^2 - 7x + 6\) and \(y = x^2 - 3x + 4\) intersect only once. Find the \(x\)-coordinate of the point of intersection. [3]
OCR MEI C1 2006 January Q12
13 marks Standard +0.3
  1. Sketch the graph of \(y = x(x - 3)^2\). [3]
  2. Show that the equation \(x(x - 3)^2 = 2\) can be expressed as \(x^3 - 6x^2 + 9x - 2 = 0\). [2]
  3. Show that \(x = 2\) is one root of this equation and find the other two roots, expressing your answers in surd form. Show the location of these roots on your sketch graph in part (i). [8]
OCR MEI C1 2006 June Q12
12 marks Moderate -0.8
You are given that \(\text{f}(x) = x^3 + 9x^2 + 20x + 12\).
  1. Show that \(x = -2\) is a root of \(\text{f}(x) = 0\). [2]
  2. Divide \(\text{f}(x)\) by \(x + 6\). [2]
  3. Express \(\text{f}(x)\) in fully factorised form. [2]
  4. Sketch the graph of \(y = \text{f}(x)\). [3]
  5. Solve the equation \(\text{f}(x) = 12\). [3]
OCR MEI C1 2009 June Q12
13 marks Moderate -0.8
  1. You are given that \(\text{f}(x) = (x + 1)(x - 2)(x - 4)\).
    1. Show that \(\text{f}(x) = x^3 - 5x^2 + 2x + 8\). [2]
    2. Sketch the graph of \(y = \text{f}(x)\). [3]
    3. The graph of \(y = \text{f}(x)\) is translated by \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\). State an equation for the resulting graph. You need not simplify your answer. Find the coordinates of the point at which the resulting graph crosses the \(y\)-axis. [3]
  2. Show that 3 is a root of \(x^3 - 5x^2 + 2x + 8 = -4\). Hence solve this equation completely, giving the other roots in surd form. [5]
OCR MEI C1 2010 June Q10
12 marks Moderate -0.3
  1. Solve, by factorising, the equation \(2x^2 - x - 3 = 0\). [3]
  2. Sketch the graph of \(y = 2x^2 - x - 3\). [3]
  3. Show that the equation \(x^2 - 5x + 10 = 0\) has no real roots. [2]
  4. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = 2x^2 - x - 3\) and \(y = x^2 - 5x + 10\). Give your answer in the form \(a \pm \sqrt{b}\). [4]
OCR MEI C1 2010 June Q12
12 marks Moderate -0.3
You are given that \(f(x) = x^3 + 6x^2 - x - 30\).
  1. Use the factor theorem to find a root of \(f(x) = 0\) and hence factorise \(f(x)\) completely. [6]
  2. Sketch the graph of \(y = f(x)\). [3]
  3. The graph of \(y = f(x)\) is translated by \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\). Show that the equation of the translated graph may be written as $$y = x^3 + 3x^2 - 10x - 24.$$ [3]
OCR MEI C1 2012 June Q11
12 marks Moderate -0.3
A cubic curve has equation \(y = f(x)\). The curve crosses the \(x\)-axis where \(x = -\frac{1}{2}\), \(-2\) and \(5\).
  1. Write down three linear factors of \(f(x)\). Hence find the equation of the curve in the form \(y = 2x^3 + ax^2 + bx + c\). [4]
  2. Sketch the graph of \(y = f(x)\). [3]
  3. The curve \(y = f(x)\) is translated by \(\begin{pmatrix} 0 \\ -8 \end{pmatrix}\). State the coordinates of the point where the translated curve intersects the \(y\)-axis. [1]
  4. The curve \(y = f(x)\) is translated by \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\) to give the curve \(y = g(x)\). Find an expression in factorised form for \(g(x)\) and state the coordinates of the point where the curve \(y = g(x)\) intersects the \(y\)-axis. [4]
OCR MEI C1 2013 June Q11
12 marks Moderate -0.8
You are given that \(\text{f}(x) = (2x - 3)(x + 2)(x + 4)\).
  1. Sketch the graph of \(y = \text{f}(x)\). [3]
  2. State the roots of \(\text{f}(x - 2) = 0\). [2]
  3. You are also given that \(\text{g}(x) = \text{f}(x) + 15\).
    1. Show that \(\text{g}(x) = 2x^3 + 9x^2 - 2x - 9\). [2]
    2. Show that \(\text{g}(1) = 0\) and hence factorise \(\text{g}(x)\) completely. [5]
Edexcel C1 Q8
10 marks Moderate -0.8
$$f(x) = 2x^2 + 3x - 2.$$
  1. Solve the equation \(f(x) = 0\). [2]
  2. Sketch the curve with equation \(y = f(x)\), showing the coordinates of any points of intersection with the coordinate axes. [2]
  3. Find the coordinates of the points where the curve with equation \(y = f(\frac{1}{2}x)\) crosses the coordinate axes. [3]
When the graph of \(y = f(x)\) is translated by 1 unit in the positive \(x\)-direction it maps onto the graph with equation \(y = ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are constants.
  1. Find the values of \(a\), \(b\) and \(c\). [3]
Edexcel C1 Q6
8 marks Moderate -0.8
  1. Sketch on the same diagram the curve with equation \(y = (x - 2)^2\) and the straight line with equation \(y = 2x - 1\). Label on your sketch the coordinates of any points where each graph meets the coordinate axes. [5]
  2. Find the set of values of \(x\) for which $$(x - 2)^2 > 2x - 1.$$ [3]
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]
Edexcel C1 Q4
6 marks Moderate -0.3
  1. Sketch on the same diagram the curves \(y = x^2 - 4x\) and \(y = -\frac{1}{x}\). [4]
  2. State, with a reason, the number of real solutions to the equation $$x^2 - 4x + \frac{1}{x} = 0.$$ [2]