Composite transformations of curvesAQA A-Level Further Maths: Subtopic test

1. 1
   The curve $C$ has equation $y=x^2+1$.

   (a)

   $C$ is rotated through $90^\circ$ anticlockwise about the origin. Find an equation of the image.

   [1 mark]

   • A$x=y^2+1$
   • B$x=-y^2-1$
   • C$y=-x^2-1$
   • D$x=1-y^2$

   (b)

   $C$ is enlarged with scale factor 2, centre the origin. Find an equation of the image.

   [1 mark]

   • A$y=2x^2+2$
   • B$y=\dfrac{x^2}{4}+1$
   • C$y=\dfrac{x^2}{2}+2$
   • D$y=8x^2+2$

   (c)

   The minimum point of $C$ is enlarged with scale factor 2, centre the origin, and its image is then translated by $\begin{pmatrix} 3 \\ -1 \end{pmatrix}$. Find the coordinates of the final position of the point.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C$ has equation $y=\mathrm{e}^x$.

   (a)

   $C$ is stretched with scale factor 3 parallel to the $y$-axis and its image is then translated by $\begin{pmatrix} 0 \\ -1 \end{pmatrix}$. Find the equation of the final image.

   [1 mark]

   • A$y=3\mathrm{e}^x-1$
   • B$y=3\mathrm{e}^x-3$
   • C$y=3\mathrm{e}^x+1$
   • D$y=\mathrm{e}^{3x}-1$

   (b)

   $C$ is rotated through $180^\circ$ about the origin. Find the equation of the image.

   [1 mark]

   • A$y=-\mathrm{e}^{x}$
   • B$y=\mathrm{e}^{-x}$
   • C$x=-\mathrm{e}^{y}$
   • D$y=-\mathrm{e}^{-x}$

   (c)

   $C$ is rotated through $180^\circ$ about the origin and the image is then translated by $\begin{pmatrix} 0 \\ 3 \end{pmatrix}$. Find the equation of the final image and write down the equation of its asymptote.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=\dfrac{1}{x-2}$.

   (a)

   $C$ is rotated through $90^\circ$ anticlockwise about the origin. Find the equation of the image, giving your answer in the form $y=\mathrm{g}(x)$.

   [3 marks]

   (b)

   $C$ is enlarged with scale factor 2, centre the origin, and its image is then translated by $\begin{pmatrix} -3 \\ 1 \end{pmatrix}$. Find the equation of the final image and the equations of its asymptotes.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=2x^2-8x+5$.

   (a)

   (i) $C$ is rotated through $180^\circ$ about the origin and its image is then translated by $\begin{pmatrix} 1 \\ 4 \end{pmatrix}$. Find the equation of the final image in the form $y=ax^2+bx+c$.
   (ii) Hence find the coordinates of the stationary point of the final image and state whether it is a maximum or a minimum.

   [6 marks]

   (b)

   Transformation $A$ is the translation $\begin{pmatrix} 0 \\ -3 \end{pmatrix}$ and transformation $B$ is the enlargement with scale factor 2, centre the origin.
   (i) Find the equation of the image of $C$ under $A$ followed by $B$.
   (ii) Find the equation of the image of $C$ under $B$ followed by $A$.
   (iii) Describe the single translation that maps the image in (i) onto the image in (ii), and explain why the order of $A$ and $B$ matters.

   [6 marks]

   Total for question 4: 12 marks

End of questions