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

1. 1
   The point $P(4,6)$ lies on the curve $y=\mathrm{f}(x)$.

   (a)

   Which point is the image of $P$ on the curve $y=\mathrm{f}(x+1)$?

   [1 mark]

   • A$(5,6)$
   • B$(4,7)$
   • C$(4,5)$
   • D$(3,6)$

   (b)

   Which point is the image of $P$ on the curve $y=\mathrm{f}\left(\frac x2\right)$?

   [1 mark]

   • A$(2,6)$
   • B$(8,6)$
   • C$(4,12)$
   • D$(4,3)$

   (c)

   The curve $y=\mathrm{f}(x)$ is transformed to $y=\mathrm{f}(-x)$. Find the coordinates of the image of $P$ and describe the transformation.

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

   $C$ is translated by $\begin{pmatrix}3\\0\end{pmatrix}$. What is the equation of the image?

   [1 mark]

   • A$y=x^2+8x+15$
   • B$y=x^2-4x+3$
   • C$y=x^2+2x-3$
   • D$y=x^2+2x+3$

   (b)

   $C$ is reflected in the $x$-axis. What is the equation of the image?

   [1 mark]

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

   (c)

   $C$ is stretched parallel to the $x$-axis with scale factor $2$. Find the equation of the image.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The ellipse $E$ has equation $\dfrac{x^2}{4}+\dfrac{y^2}{9}=1$.

   (a)

   $E$ is reflected in the line $y=x$. Find the equation of the image, and the coordinates of the points where the image meets the $x$-axis.

   [3 marks]

   (b)

   $E$ is stretched parallel to the $y$-axis with scale factor $\frac13$. Find the equation of the image, and the coordinates of the points where the image meets the $y$-axis.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=x^2-2x-3$.

   (a)

   Find the equation of the image of $C$ under each of these single transformations.
   (i) Reflection in the $y$-axis.
   (ii) Stretch parallel to the $y$-axis with scale factor $3$.
   (iii) Translation by $\begin{pmatrix}2\\0\end{pmatrix}$.

   [6 marks]

   (b)

   Describe fully the single transformation that maps $C$ onto each of these curves.
   (i) $y=x^2-2x+1$
   (ii) $y=-x^2+2x+3$
   (iii) $y=\frac{x^2}{4}-x-3$

   [6 marks]

   Total for question 4: 12 marks

End of questions