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

An ellipse \(E\) has equation $$\frac{x^2}{16} + \frac{y^2}{9} = 1$$

1. Sketch the ellipse \(E\), showing the values of the intercepts on the coordinate axes. [2 marks]
2. Given that the line with equation \(y = x + k\) intersects the ellipse \(E\) at two distinct points, show that \(-5 < k < 5\). [5 marks]
3. The ellipse \(E\) is translated by the vector \(\begin{bmatrix} a \\ b \end{bmatrix}\) to form another ellipse whose equation is \(9x^2 + 16y^2 + 18x - 64y = c\). Find the values of the constants \(a\), \(b\) and \(c\). [5 marks]
4. Hence find an equation for each of the two tangents to the ellipse \(9x^2 + 16y^2 + 18x - 64y = c\) that are parallel to the line \(y = x\). [3 marks]

A parabola with equation \(y^2 = 4ax\), where \(a\) is a constant, is translated by the vector \(\begin{bmatrix} 2 \\ 3 \end{bmatrix}\) to give the curve \(C\). The curve \(C\) passes through the point \((4, 7)\).

1. Show that \(a = 2\). [3 marks]
2. Find the values of \(k\) for which the line \(ky = x\) does not meet the curve \(C\). [6 marks]

1. Given that \(y = \ln [t + \sqrt{(1 + t^2)}]\), show that \(\frac{dy}{dt} = \frac{1}{\sqrt{(1+t^2)}}\). [3]

The curve \(C\) has parametric equations $$x = \frac{1}{\sqrt{(1+t^2)}}, \quad y = \ln [t + \sqrt{(1 + t^2)}], \quad t \in \mathbb{R}.$$ A student was asked to prove that, for \(t > 0\), the gradient of the tangent to \(C\) is negative. The attempted proof was as follows: $$y = \ln \left(t + \frac{1}{x}\right)$$ $$= \ln \left(\frac{tx + 1}{x}\right)$$ $$= \ln (tx + 1) - \ln x$$ $$\therefore \frac{dy}{dx} = \frac{t}{tx + 1} - \frac{1}{x}$$ $$= \frac{\frac{t}{x}}{t + \frac{1}{x}} - \frac{1}{x}$$ $$= \frac{t\sqrt{(1+t^2)}}{t + \sqrt{(1+t^2)}} - \sqrt{(1 + t^2)}$$ $$= -\frac{(1+t^2)}{t + \sqrt{(1+t^2)}}$$ As \((1 + t^2) > 0\), and \(t + \sqrt{(1 + t^2)} > 0\) for \(t > 0\), \(\frac{dy}{dx} < 0\) for \(t > 0\).

1. 1. Identify the error in this attempt.
   2. Give a correct version of the proof. [6]
2. Prove that \(\ln [-t + \sqrt{(1 + t^2)}] = -\ln [t + \sqrt{(1 + t^2)}]\). [3]
3. Deduce that \(C\) is symmetric about the \(x\)-axis and sketch the graph of \(C\). [3]

$$f(x) = \frac{ax + b}{x + 2}; \quad x \in \mathbb{R}, x \neq -2,$$ where \(a\) and \(b\) are constants and \(b > 0\).

1. Find \(f^{-1}(x)\). [2]
2. Hence, or otherwise, find the value of \(a\) so that \(f(x) = x\). [2]

The curve \(C\) has equation \(y = f(x)\) and \(f(x)\) satisfies \(f(x) = x\).

1. On separate axes sketch
   1. \(y = f(x)\), [3]
   2. \(y = f(x - 2) + 2\). [3]

On each sketch you should indicate the equations of any asymptotes and the coordinates, in terms of \(b\), of any intersections with the axes. The normal to \(C\) at the point \(P\) has equation \(y = 4x - 39\). The normal to \(C\) at the point \(Q\) has equation \(y = 4x + k\), where \(k\) is a constant.

1. By considering the images of the normals to \(C\) on the curve with equation \(y = f(x - 2) + 2\), or otherwise, find the value of \(k\). [5]

A circle with centre \(C\) has equation \(x^2 + y^2 - 6x + 4y + 4 = 0\).

1. Find
   1. the coordinates of \(C\), [2]
   2. the radius of the circle. [1]
2. Determine the set of values of \(k\) for which the line \(y = kx - 3\) does not intersect or touch the circle. [5]

A sequence of transformations maps the curve \(y = e^x\) to the curve \(y = e^{2x+3}\). Give details of these transformations. [3]

1. Give full details of the single transformation that transforms the graph of \(y = x^3\) to the graph of \(y = x^3 - 8\). [2]

The function f is defined by \(\mathrm{f}(x) = x^3 - 8\).

1. Find an expression for \(\mathrm{f}^{-1}(x)\). [2]
2. State how the graphs of \(y = \mathrm{f}(x)\) and \(y = \mathrm{f}^{-1}(x)\) are related geometrically. [1]

The cubic polynomial \(\text{f}(x)\) is defined by \(\text{f}(x) = x^3 + px + q\), where \(p\) and \(q\) are constants.

1. 1. Given that \(\text{f}'(2) = 13\), find the value of \(p\). [2]
   2. Given also that \((x - 2)\) is a factor of \(\text{f}(x)\), find the value of \(q\). [2]
   The curve \(y = \text{f}(x)\) is translated by the vector \(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\).
2. Using the values from part (a), determine the equation of the curve after it has been translated. Give your answer in the form \(y = x^3 + ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are integers to be found. [4]

Curve C has equation \(y = x^2\) C is translated by vector \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\) to give curve \(C_1\) Line L has equation \(y = x\) L is stretched by scale factor 2 parallel to the \(x\)-axis to give line \(L_1\) Find the exact distance between the two intersection points of \(C_1\) and \(L_1\) [6 marks]

The graph of the equation \(y = \frac{1}{x}\) is translated by the vector \(\begin{bmatrix}3\\0\end{bmatrix}\)

1. Write down the equation of the transformed graph. [1 mark]
2. State the equations of the asymptotes of the transformed graph. [2 marks]

The curve \(y = \sqrt{x}\) is translated onto the curve \(y = \sqrt{x + 4}\) The translation is described by a vector. Find this vector. Circle your answer. [1 mark] \(\begin{bmatrix} 4 \\ 0 \end{bmatrix}\) \(\begin{bmatrix} -4 \\ 0 \end{bmatrix}\) \(\begin{bmatrix} 0 \\ 4 \end{bmatrix}\) \(\begin{bmatrix} 0 \\ -4 \end{bmatrix}\)

Figure 1 shows \(y = f(x)\). \includegraphics{figure_1} Which figure below shows \(y = f(2x)\)? Tick one box. \includegraphics{figure_2} \quad \includegraphics{figure_3} \quad \includegraphics{figure_4} \quad \includegraphics{figure_5} [1 mark]

The curve C has equation \(y = f(x)\) C has a maximum point at P with coordinates \((a, 2b)\) as shown in the diagram below. \includegraphics{figure_7}

1. C is mapped by a single transformation onto curve \(C_1\) with equation \(y = f(x + 2)\) State the coordinates of the maximum point on curve \(C_1\) [1 mark]
2. C is mapped by a single transformation onto curve \(C_2\) with equation \(y = 4f(x)\) State the coordinates of the maximum point on curve \(C_2\) [1 mark]
3. C is mapped by a stretch in the \(x\)-direction onto curve \(C_3\) with equation \(y = f(3x)\) State the scale factor of the stretch. [1 mark]

Curve \(C\) has equation \(y = 8 \sin x\)

1. Curve \(C\) is transformed onto curve \(C_1\) by a translation of vector \(\begin{pmatrix} 0 \\ 4 \end{pmatrix}\) Find the equation of \(C_1\) [1 mark]
2. Curve \(C\) is transformed onto curve \(C_2\) by a stretch of scale factor 4 in the \(y\) direction. Find the equation of \(C_2\) [1 mark]
3. Curve \(C\) is transformed onto curve \(C_3\) by a stretch of scale factor 2 in the \(x\) direction. Find the equation of \(C_3\) [1 mark]

The graph of \(y = f(x)\) is shown in Figure 1. \includegraphics{figure_1} State the equation of the graph shown in Figure 2. \includegraphics{figure_2} Circle your answer. [1 mark] \(y = f(2x)\) \quad\quad \(y = f\left(\frac{x}{2}\right)\) \quad\quad \(y = 2f(x)\) \quad\quad \(y = \frac{1}{2}f(x)\)

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}

Identify the graph of \(y = 1 - |x + 2|\) from the options below. Tick (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_1}

The graph of \(y = f(x)\) is shown below. \includegraphics{figure_6}

1. Sketch the graph of \(y = f(-x)\) [2 marks]
2. Sketch the graph of \(y = 2f(x) - 4\) [2 marks]
3. Sketch the graph of \(y = f'(x)\) [3 marks]

A function f is defined for all real values of \(x\) as $$f(x) = x^4 + 5x^3$$ The function has exactly two stationary points when \(x = 0\) and \(x = -\frac{15}{4}\)

1. 1. Find \(f''(x)\) [2 marks]
   2. Determine the nature of the stationary points. Fully justify your answer. [4 marks]
2. State the range of values of \(x\) for which $$f(x) = x^4 + 5x^3$$ is an increasing function. [1 mark]
3. A second function g is defined for all real values of \(x\) as $$g(x) = x^4 - 5x^3$$
   1. State the single transformation which maps f onto g. [1 mark]
   2. State the range of values of \(x\) for which g is an increasing function. [1 mark]

Given that \(\text{f}(x) = x^3\) and \(\text{g}(x) = 2x^3 - 1\), describe a sequence of two transformations which maps the curve \(y = \text{f}(x)\) onto the curve \(y = \text{g}(x)\). [4]

The mean value of the function \(\mathbf{f}\) over the interval \(1 \leq x \leq 5\) is \(m\). The graph of \(y = \mathbf{g}(x)\) is a reflection in the \(x\)-axis of \(y = \mathbf{f}(x)\). The graph of \(y = \mathbf{h}(x)\) is a translation of \(y = \mathbf{g}(x)\) by \(\begin{bmatrix} 3 \\ 7 \end{bmatrix}\) Determine, in terms of \(m\), the mean value of the function \(\mathbf{h}\) over the interval \(4 \leq x \leq 8\) [2 marks]