1.02n Sketch curves: simple equations including polynomials

487 questions

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OCR FP2 2010 January Q8
10 marks Standard +0.3
The equation of a curve is $$y = \frac{kx}{(x-1)^2},$$ where \(k\) is a positive constant.
  1. Write down the equations of the asymptotes of the curve. [2]
  2. Show that \(y \geq -\frac{1}{4}k\). [4]
  3. Show that the \(x\)-coordinate of the stationary point of the curve is independent of \(k\), and sketch the curve. [4]
Edexcel AEA 2002 June Q6
17 marks Hard +2.3
\includegraphics{figure_2} Figure 2 shows a sketch of part of two curves \(C_1\) and \(C_2\) for \(y \geq 0\). The equation of \(C_1\) is \(y = m_1 - x^{n_1}\) and the equation of \(C_2\) is \(y = m_2 - x^{n_2}\), where \(m_1\), \(m_2\), \(n_1\) and \(n_2\) are positive integers with \(m_2 > m_1\). Both \(C_1\) and \(C_2\) are symmetric about the line \(x = 0\) and they both pass through the points \((3, 0)\) and \((-3, 0)\). Given that \(n_1 + n_2 = 12\), find
  1. the possible values of \(n_1\) and \(n_2\), [4]
  2. the exact value of the smallest possible area between \(C_1\) and \(C_2\), simplifying your answer, [8]
  3. the largest value of \(x\) for which the gradients of the two curves can be the same. Leave your answer in surd form. [5]
Edexcel AEA 2004 June Q6
17 marks Challenging +1.8
$$f(x) = x - [x], \quad x \geq 0$$ where \([x]\) is the largest integer \(\leq x\). For example, \(f(3.7) = 3.7 - 3 = 0.7\); \(f(3) = 3 - 3 = 0\).
  1. Sketch the graph of \(y = f(x)\) for \(0 \leq x < 4\). [3]
  2. Find the value of \(p\) for which \(\int_2^p f(x) dx = 0.18\). [3]
Given that $$g(x) = \frac{1}{1+kx}, \quad x \geq 0, \quad k > 0,$$ and that \(x_0 = \frac{1}{2}\) is a root of the equation \(f(x) = g(x)\),
  1. find the value of \(k\). [2]
  2. Add a sketch of the graph of \(y = g(x)\) to your answer to part \((a)\). [1]
The root of \(f(x) = g(x)\) in the interval \(n < x < n + 1\) is \(x_n\), where \(n\) is an integer.
  1. Prove that $$2 x_n^2 - (2n - 1)x_n - (n + 1) = 0.$$ [4]
  2. Find the smallest value of \(n\) for which \(x_n - n < 0.05\). [4]
AQA AS Paper 1 2019 June Q5
5 marks Moderate -0.8
  1. Sketch the curve \(y = g(x)\) where $$g(x) = (x + 2)(x - 1)^2$$ [3 marks]
  2. Hence, solve \(g(x) \leq 0\) [2 marks]
AQA AS Paper 1 2021 June Q5
6 marks Moderate -0.3
  1. Sketch the curve $$y = (x - a)^2(3 - x) \quad \text{where } 0 < a < 3$$ indicating the coordinates of the points where the curve and the axes meet. [4 marks] \includegraphics{figure_5}
  2. Hence, solve $$(x - a)^2(3 - x) > 0$$ giving your answer in set notation form. [2 marks]
AQA Paper 1 2019 June Q7
11 marks Standard +0.3
  1. By sketching the graphs of \(y = \frac{1}{x}\) and \(y = \sec 2x\) on the axes below, show that the equation $$\frac{1}{x} = \sec 2x$$ has exactly one solution for \(x > 0\) [3 marks] \includegraphics{figure_7a}
  2. By considering a suitable change of sign, show that the solution to the equation lies between 0.4 and 0.6 [2 marks]
  3. Show that the equation can be rearranged to give $$x = \frac{1}{2}\cos^{-1}x$$ [2 marks]
    1. Use the iterative formula $$x_{n+1} = \frac{1}{2}\cos^{-1}x_n$$ with \(x_1 = 0.4\), to find \(x_2\), \(x_3\) and \(x_4\), giving your answers to four decimal places. [2 marks]
    2. On the graph below, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x_2\), \(x_3\) and \(x_4\). [2 marks] \includegraphics{figure_7d}
AQA Paper 1 2024 June Q11
5 marks Standard +0.3
It is given that $$f(x) = x(x - a)(x - 6)$$ where \(0 < a < 6\)
  1. Sketch the graph of \(y = f(x)\) on the axes below. [3 marks] \includegraphics{figure_11a}
  2. Sketch the graph of \(y = f(-2x)\) on the axes below. [2 marks] \includegraphics{figure_11b}
AQA Paper 2 2018 June Q4
6 marks Standard +0.3
A curve, C, has equation \(y = x^2 - 6x + k\), where \(k\) is a constant. The equation \(x^2 - 6x + k = 0\) has two distinct positive roots.
  1. Sketch C on the axes below. [2 marks]
  2. Find the range of possible values for \(k\). Fully justify your answer. [4 marks]
AQA Paper 2 2019 June Q7
10 marks Challenging +1.2
  1. Sketch the graph of any cubic function that has both three distinct real roots and a positive coefficient of \(x^3\) [2 marks]
  2. The function f(x) is defined by $$f(x) = x^3 + 3px^2 + q$$ where \(p\) and \(q\) are constants and \(p > 0\)
    1. Show that there is a turning point where the curve crosses the \(y\)-axis. [3 marks]
    2. The equation \(f(x) = 0\) has three distinct real roots. By considering the positions of the turning points find, in terms of \(p\), the range of possible values of \(q\). [5 marks]
AQA Paper 2 Specimen Q6
5 marks Moderate -0.8
A curve \(C\) has equation \(y = x^2 - 4x + k\), where \(k\) is a constant. It crosses the \(x\)-axis at the points \((2 + \sqrt{5}, 0)\) and \((2 - \sqrt{5}, 0)\)
  1. Find the value of \(k\). [2 marks]
  2. Sketch the curve \(C\), labelling the exact values of all intersections with the axes. [3 marks]
AQA Paper 3 2023 June Q6
9 marks Standard +0.3
  1. Sketch the curve with equation $$y = x^2(2x + a)$$ where \(a > 0\) [3 marks] \includegraphics{figure_6a}
  2. The polynomial \(p(x)\) is given by $$p(x) = x^2(2x + a) + 36$$
    1. It is given that \(x + 3\) is a factor of \(p(x)\) Use the factor theorem to show \(a = 2\) [2 marks]
    2. State the transformation which maps the curve with equation $$y = x^2(2x + 2)$$ onto the curve with equation $$y = x^2(2x + 2) + 36$$ [2 marks]
    3. The polynomial \(x^2(2x + 2) + 36\) can be written as \((x + 3)(2x^2 + bx + c)\) Without finding the values of \(b\) and \(c\), use your answers to parts (a) and (b)(ii) to explain why $$b^2 < 8c$$ [2 marks]
Edexcel AS Paper 1 Specimen Q13
7 marks Standard +0.3
  1. Factorise completely \(x^3 + 10x^2 + 25x\) [2]
  2. Sketch the curve with equation $$y = x^3 + 10x^2 + 25x$$ showing the coordinates of the points at which the curve cuts or touches the \(x\)-axis. [2]
The point with coordinates \((-3, 0)\) lies on the curve with equation $$y = (x + a)^3 + 10(x + a)^2 + 25(x + a)$$ where \(a\) is a constant.
  1. Find the two possible values of \(a\). [3]
OCR PURE Q6
7 marks Easy -1.2
Sketch the following curves.
  1. \(y = \frac{2}{x}\) [2]
  2. \(y = x^3 - 6x^2 + 9x\) [5]
OCR MEI Paper 2 2022 June Q16
15 marks Standard +0.3
The equation of a curve is $$y = 6x^4 + 8x^3 - 21x^2 + 12x - 6.$$
  1. In this question you must show detailed reasoning. Determine
    [12]
  2. On the axes in the Printed Answer Booklet, sketch the curve whose equation is $$y = 6x^4 + 8x^3 - 21x^2 + 12x - 6.$$ [3]
AQA Further AS Paper 1 2018 June Q13
9 marks Challenging +1.2
The graph of the rational function \(y = f(x)\) intersects the \(x\)-axis exactly once at \((-3, 0)\) The graph has exactly two asymptotes, \(y = 2\) and \(x = -1\)
  1. Find \(f(x)\) [2 marks]
  2. Sketch the graph of the function. [3 marks] \includegraphics{figure_13b}
  3. Find the range of values of \(x\) for which \(f(x) \leq 5\) [4 marks]
AQA Further AS Paper 1 2020 June Q10
8 marks Standard +0.3
  1. Show that the equation $$y = \frac{3x - 5}{2x + 4}$$ can be written in the form $$(x + a)(y + b) = c$$ where \(a\), \(b\) and \(c\) are integers to be found. [3 marks]
  2. Write down the equations of the asymptotes of the graph of $$y = \frac{3x - 5}{2x + 4}$$ [2 marks]
  3. Sketch, on the axes provided, the graph of $$y = \frac{3x - 5}{2x + 4}$$ \includegraphics{figure_10} [3 marks]
AQA Further Paper 1 Specimen Q8
5 marks Standard +0.3
A curve has equation $$y = \frac{5 - 4x}{1 + x}$$
  1. Sketch the curve. [4 marks]
  2. Hence sketch the graph of \(y = \left|\frac{5 - 4x}{1 + x}\right|\). [1 mark]
WJEC Unit 1 2019 June Q13
11 marks Moderate -0.8
A curve \(C\) has equation \(y = \frac{1}{9}x^3 - kx + 5\). A point \(Q\) lies on \(C\) and is such that the tangent to \(C\) at \(Q\) has gradient \(-9\). The \(x\)-coordinate of \(Q\) is \(3\).
  1. Show that \(k = 12\). [3]
  2. Find the coordinates of each of the stationary points of \(C\) and determine their nature. [6]
  3. Sketch the curve \(C\), clearly labelling the stationary points and the point where the curve crosses the \(y\)-axis. [2]
WJEC Unit 1 2022 June Q5
9 marks Moderate -0.8
The curve \(C_1\) has equation \(y = -x^2 + 2x + 3\) and the curve \(C_2\) has equation \(y = x^2 - x - 6\). The two curves intersect at the points \(A\) and \(B\).
  1. Determine the coordinates of \(A\) and \(B\). [4]
  2. On the same set of axes, sketch the graphs of \(C_1\) and \(C_2\). Clearly label the points where the two curves intersect. [3]
  3. In the diagram drawn in part (b), shade the region satisfying the following inequalities: [2] $$x > 0,$$ $$y < -x^2 + 2x + 3,$$ $$y > x^2 - x - 6.$$
WJEC Unit 1 2024 June Q16
10 marks Moderate -0.8
  1. Find the range of values of \(k\) for which the quadratic equation \(x^2 - kx + 4 = 0\) has no real roots. [4]
  2. Determine the coordinates of the points of intersection of the graphs of \(y = x^2 - 3x + 4\) and \(y = x + 16\). [4]
  3. Using the information obtained in parts (a) and (b), sketch the graphs of \(y = x^2 - 3x + 4\) and \(y = x + 16\) on the same set of axes. [2]
WJEC Unit 3 Specimen Q3
8 marks Moderate -0.3
  1. Sketch the graph of \(y = x^2 + 6x + 13\), identifying the stationary point. [2]
  2. The function \(f\) is defined by \(f(x) = x^2 + 6x + 13\) with domain \((a,b)\).
    1. Explain why \(f^{-1}\) does not exist when \(a = -10\) and \(b = 10\). [1]
    2. Write down a value of \(a\) and a value of \(b\) for which the inverse of \(f\) does exist and derive an expression for \(f^{-1}(x)\). [5]
SPS SPS SM 2020 October Q7
11 marks Moderate -0.3
  1. Sketch the curves \(y = \frac{3}{x^2}\) and \(y = x^2 - 2\) on the axes provided below. \includegraphics{figure_1} [3]
  2. In this question you must show detailed reasoning. Find the exact coordinates of the points of interception of the curves \(y = \frac{3}{x^2}\) and \(y = x^2 - 2\). [6]
  3. Hence, solve the inequality \(\frac{3}{x^2} \leq x^2 - 2\), giving your answer in interval notation. [2]
SPS SPS SM 2022 October Q2
6 marks Easy -1.2
  1. Complete the square for \(1 - 4x - x^2\) [3]
  2. Sketch the curve \(y = 1 - 4x - x^2\), including the coordinates of any maximum or minimum points and the y intercept only. [3]
SPS SPS SM 2022 February Q4
8 marks Easy -1.3
  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]
SPS SPS SM 2022 October Q2
5 marks Easy -1.3
A curve \(C\) has equation \(y = f(x)\) where $$f(x) = -3x^2 + 12x + 8$$
  1. Write \(f(x)\) in the form $$a(x + b)^2 + c$$ where \(a\), \(b\) and \(c\) are constants to be found. [3]
The curve \(C\) has a maximum turning point at \(M\).
  1. Find the coordinates of \(M\). [2]