1.02w Graph transformations: simple transformations of f(x)

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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 Q7
9 marks Standard +0.3
  1. Describe fully a single transformation that maps the graph of \(y = \frac{1}{x}\) onto the graph of \(y = \frac{3}{x}\). [2]
  2. Sketch the graph of \(y = \frac{3}{x}\) and write down the equations of any asymptotes. [3]
  3. Find the values of the constant \(c\) for which the straight line \(y = c - 3x\) is a tangent to the curve \(y = \frac{3}{x}\). [4]
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 Q8
10 marks Moderate -0.3
  1. Describe fully the single transformation that maps the graph of \(y = \text{f}(x)\) onto the graph of \(y = \text{f}(x - 1)\). [2]
  2. Showing the coordinates of any points of intersection with the coordinate axes and the equations of any asymptotes, sketch the graph of \(y = \frac{1}{x-1}\). [3]
  3. Find the \(x\)-coordinates of any points where the graph of \(y = \frac{1}{x-1}\) intersects the graph of \(y = 2 + \frac{1}{x}\). Give your answers in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are rational. [5]
OCR C1 Q7
10 marks Standard +0.3
  1. Describe fully the single transformation that maps the graph of \(y = f(x)\) onto the graph of \(y = f(x - 1)\). [2]
  2. Showing the coordinates of any points of intersection with the coordinate axes and the equations of any asymptotes, sketch the graph of \(y = \frac{1}{x-1}\). [3]
  3. Find the \(x\)-coordinates of any points where the graph of \(y = \frac{1}{x-1}\) intersects the graph of \(y = 2 + \frac{1}{x}\). Give your answers in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are rational. [5]
OCR C1 Q5
6 marks Moderate -0.8
\includegraphics{figure_5} The diagram shows a sketch of the curve with equation \(y = f(x)\). The curve has a maximum at \((-3, 4)\) and a minimum at \((1, -2)\). Showing the coordinates of any turning points, sketch on separate diagrams the curves with equations
  1. \(y = 2f(x)\), [3]
  2. \(y = -f(x)\). [3]
OCR C1 Q5
8 marks Moderate -0.3
$$f(x) = x^2 - 10x + 17.$$
  1. Express \(f(x)\) in the form \(a(x + b)^2 + c\). [3]
  2. State the coordinates of the minimum point of the curve \(y = f(x)\). [1]
  3. Deduce the coordinates of the minimum point of each of the following curves:
    1. \(y = f(x) + 4\), [2]
    2. \(y = f(2x)\). [2]
OCR C1 Q4
6 marks Moderate -0.8
The curve \(C\) has the equation \(y = (x - a)^2\) where \(a\) is a constant. Given that $$\frac{dy}{dx} = 2x - 6,$$ \begin{enumerate}[label=(\roman*)] \item find the value of \(a\), [4] \item describe fully a single transformation that would map \(C\) onto the graph of \(y = x^2\). [2]
OCR MEI C1 Q6
11 marks Moderate -0.3
  1. \includegraphics{figure_1} Fig. 10 shows a sketch of the graph of \(y = \frac{1}{x}\). Sketch the graph of \(y = \frac{1}{x-2}\), showing clearly the coordinates of any points where it crosses the axes. [3]
  2. Find the value of \(x\) for which \(\frac{1}{x-2} = 5\). [2]
  3. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = x\) and \(y = \frac{1}{x-2}\). Give your answers in the form \(a \pm \sqrt{b}\). Show the position of these points on your graph in part (i). [6]
OCR MEI C1 Q4
12 marks Moderate -0.8
Answer the whole of this question on the insert provided. The insert shows the graph of \(y = \frac{1}{x}\), \(x \neq 0\).
  1. Use the graph to find approximate roots of the equation \(\frac{1}{x} = 2x + 3\), showing your method clearly. [3]
  2. Rearrange the equation \(\frac{1}{x} = 2x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac{p \pm \sqrt{q}}{r}\). [5]
  3. Draw the graph of \(y = \frac{1}{x} + 2\), \(x \neq 0\), on the grid used for part (i). [2]
  4. Write down the values of \(x\) which satisfy the equation \(\frac{1}{x} + 2 = 2x + 3\). [2]
OCR MEI C1 Q6
13 marks Moderate -0.8
\includegraphics{figure_6} Fig. 11 shows a sketch of the curve with equation \(y = (x - 4)^2 - 3\).
  1. Write down the equation of the line of symmetry of the curve and the coordinates of the minimum point. [2]
  2. Find the coordinates of the points of intersection of the curve with the \(x\)-axis and the \(y\)-axis, using surds where necessary. [4]
  3. The curve is translated by \(\begin{pmatrix} 2 \\ 0 \end{pmatrix}\). Show that the equation of the translated curve may be written as \(y = x^2 - 12x + 33\). [2]
  4. Show that the line \(y = 8 - 2x\) meets the curve \(y = x^2 - 12x + 33\) at just one point, and find the coordinates of this point. [5]
OCR MEI C1 Q7
4 marks Easy -1.2
  1. Describe fully the transformation which maps the curve \(y = x^2\) onto the curve \(y = (x + 4)^2\). [2]
  2. Sketch the graph of \(y = x^2 - 4\). [2]
AQA C2 2009 June Q4
6 marks Moderate -0.3
  1. Use the trapezium rule with four ordinates (three strips) to find an approximate value for \(\int_0^6 \sqrt{x^3 + 1} dx\), giving your answer to four significant figures. [4]
  2. The curve with equation \(y = \sqrt{x^3 + 1}\) is stretched parallel to the \(x\)-axis with scale factor \(\frac{1}{2}\) to give the curve with equation \(y = f(x)\). Write down an expression for \(f(x)\). [2]
OCR MEI C2 2010 January Q5
4 marks Easy -1.2
\includegraphics{figure_5} Fig. 5 shows a sketch of the graph of \(y = f(x)\). On separate diagrams, sketch the graphs of the following, showing clearly the coordinates of the points corresponding to P, Q and R.
  1. \(y = f(2x)\) [2]
  2. \(y = \frac{1}{2}f(x)\) [2]
OCR MEI C2 2013 January Q3
4 marks Moderate -0.8
  1. The point P\((4, -2)\) lies on the curve \(y = f(x)\). Find the coordinates of the image of P when the curve is transformed to \(y = f(5x)\). [2]
  2. Describe fully a single transformation which maps the curve \(y = \sin x^2\) onto the curve \(y = \sin(x - 90)^2\). [2]
OCR MEI C2 2008 June Q3
2 marks Easy -1.2
State the transformation which maps the graph of \(y = x^2 + 5\) onto the graph of \(y = 3x^2 + 15\). [2]
OCR MEI C2 2008 June Q11
12 marks Moderate -0.3
\includegraphics{figure_11} Fig. 11 shows a sketch of the cubic curve \(y = \text{f}(x)\). The values of \(x\) where it crosses the \(x\)-axis are \(-5\), \(-2\) and \(2\), and it crosses the \(y\)-axis at \((0, -20)\).
  1. Express f(\(x\)) in factorised form. [2]
  2. Show that the equation of the curve may be written as \(y = x^3 + 5x^2 - 4x - 20\). [2]
  3. Use calculus to show that, correct to 1 decimal place, the \(x\)-coordinate of the minimum point on the curve is 0.4. Find also the coordinates of the maximum point on the curve, giving your answers correct to 1 decimal place. [6]
  4. State, correct to 1 decimal place, the coordinates of the maximum point on the curve \(y = \text{f}(2x)\). [2]
OCR MEI C2 2010 June Q4
4 marks Moderate -0.8
In this question, \(f(x) = x^2 - 5x\). Fig. 4 shows a sketch of the graph of \(y = f(x)\). \includegraphics{figure_4} On separate diagrams, sketch the curves \(y = f(2x)\) and \(y = 3f(x)\), labelling the coordinates of their intersections with the axes and their turning points. [4]
OCR MEI C2 2013 June Q8
4 marks Easy -1.3
Fig. 8 shows the graph of \(y = g(x)\). \includegraphics{figure_8} Draw the graph of
  1. \(y = g(2x)\), [2]
  2. \(y = 3g(x)\). [2]
OCR MEI C2 2014 June Q4
4 marks Moderate -0.8
The point R\((6, -3)\) is on the curve \(y = f(x)\).
  1. Find the coordinates of the image of R when the curve is transformed to \(y = \frac{1}{2}f(x)\). [2]
  2. Find the coordinates of the image of R when the curve is transformed to \(y = f(3x)\). [2]
OCR MEI C2 2016 June Q5
4 marks Easy -1.3
  1. Fig. 5 shows the graph of a sine function. \includegraphics{figure_5} State the equation of this curve. [2]
  2. Sketch the graph of \(y = \sin x - 3\) for \(0° \leq x \leq 450°\). [2]
Edexcel C2 Q5
9 marks Moderate -0.3
  1. Describe fully a single transformation that maps the graph of \(y = 3^x\) onto the graph of \(y = (\frac{1}{3})^x\). [1]
  2. Sketch on the same diagram the curves \(y = (\frac{1}{3})^x\) and \(y = 2(3^x)\), showing the coordinates of any points where each curve crosses the coordinate axes. [3]
The curves \(y = (\frac{1}{3})^x\) and \(y = 2(3^x)\) intersect at the point \(P\).
  1. Find the \(x\)-coordinate of \(P\) to 2 decimal places and show that the \(y\)-coordinate of \(P\) is \(\sqrt{2}\). [5]
OCR MEI C2 Q1
13 marks Moderate -0.3
The gradient of a curve is given by \(\frac{dy}{dx} = 4x + 3\). The curve passes through the point \((2, 9)\).
  1. Find the equation of the tangent to the curve at the point \((2, 9)\). [3]
  2. Find the equation of the curve and the coordinates of its points of intersection with the \(x\)-axis. Find also the coordinates of the minimum point of this curve. [7]
  3. Find the equation of the curve after it has been stretched parallel to the \(x\)-axis with scale factor \(\frac{1}{2}\). Write down the coordinates of the minimum point of the transformed curve. [3]
OCR MEI C2 Q3
5 marks Easy -1.3
  1. On the same axes, sketch the graphs of \(y = \cos x\) and \(y = \cos 2x\) for values of \(x\) from \(0\) to \(2\pi\). [3]
  2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = 3 \cos x\). [2]