2.8 Transformations of graphsIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   The graph of $y=f(x)$, where $f(x)=x^2+4$, is translated by the vector $\begin{pmatrix}3\\ -2\end{pmatrix}$ to give the graph of $y=g(x)$.

   (a)

   The point $(1,5)$ lies on the graph of $y=f(x)$. Which point lies on the graph of $y=g(x)$?

   [1 mark]

   • A$(-2,7)$
   • B$(4,7)$
   • C$(4,3)$
   • D$(-2,3)$

   (b)

   Which expression gives $g(x)$ in terms of $f$?

   [1 mark]

   • A$f(x+3)-2$
   • B$f(x-3)+2$
   • C$f(x+3)+2$
   • D$f(x-3)-2$

   (c)

   Find $g(x)$ in the form $ax^2+bx+c$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The point $P(4,6)$ lies on the graph of $y=f(x)$.

   (a)

   Find the image of $P$ on the graph of $y=f(-x)$.

   [1 mark]

   • A$(-4,6)$
   • B$(4,-6)$
   • C$(-4,-6)$
   • D$(6,4)$

   (b)

   Find the image of $P$ on the graph of $y=f(3x)$.

   [1 mark]

   • A$(12,6)$
   • B$\left(\frac43,6\right)$
   • C$(4,18)$
   • D$(4,2)$

   (c)

   The graph of $y=f(x)$ is stretched vertically with scale factor 3 and the resulting graph is then reflected in the $x$-axis. Write down the equation of the final graph and the coordinates of the image of $P$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The graph of $y=\sin x$ ($x$ in radians) is transformed to give the graph of $y=g(x)$, where $g(x)=4\sin 2x+1$.

   (a)

   Describe fully a sequence of three transformations that maps the graph of $y=\sin x$ onto the graph of $y=g(x)$.

   [3 marks]

   (b)

   (i) Write down the range of $g$.
   (ii) The graph of $y=g(x)$ is reflected in the $x$-axis and then translated by the vector $\begin{pmatrix}\frac{\pi}{4}\\ 0\end{pmatrix}$. Find the equation of the resulting graph.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The population $P$ (in thousands) of an insect colony $t$ weeks after it was introduced is modelled by $P=f(t)$, where $f(t)=2^{t}$. The graph of $y=f(t)$ is transformed to model other colonies.

   (a)

   A second colony is modelled by $g(t)=3f(t-2)$.
   (i) Describe the two transformations that map the graph of $y=f(t)$ onto the graph of $y=g(t)$.
   (ii) Show that $g(t)=\frac34\times2^{t}$.
   (iii) Hence find the time at which the second colony first reaches 24 thousand.

   [6 marks]

   (b)

   Anna applies a vertical stretch with scale factor 3 to the graph of $y=f(t)$ and then translates the result by the vector $\begin{pmatrix}0\\ -2\end{pmatrix}$. Ben applies the same two transformations in the opposite order.
   (i) Find the equation of Anna's graph.
   (ii) Find the equation of Ben's graph.
   (iii) Find the $y$-intercept of each graph and hence explain why the order of the transformations matters.

   [6 marks]

   Total for question 4: 12 marks

End of questions