1. 1
   The graph of $y = f(x)$ passes through the point $P(2, -3)$.

   (a)

   Find the coordinates of the image of $P$ on the graph of $y = f(x - 4) + 1$.

   [1 mark]

   • A$(-2, -2)$
   • B$(6, -4)$
   • C$(6, -2)$
   • D$(-2, -4)$

   (b)

   Find the coordinates of the image of $P$ on the graph of $y = -2f(x)$.

   [1 mark]

   • A$(2, 6)$
   • B$(2, -6)$
   • C$\left(2, \frac{3}{2}\right)$
   • D$(4, -3)$

   (c)

   Find the coordinates of the image of $P$ on the graph of $y = 3f\left(\frac{1}{2}x\right)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $f(x) = x^{2}$. The graph of $y = f(x)$ is reflected in the $x$-axis and then translated 3 units in the positive $x$-direction and 4 units in the positive $y$-direction, to give the graph of $y = g(x)$.

   (a)

   Find an expression for $g(x)$.

   [1 mark]

   • A$-(x + 3)^{2} + 4$
   • B$-(x - 3)^{2} + 4$
   • C$-(x - 3)^{2} - 4$
   • D$(x - 3)^{2} + 4$

   (b)

   Write down the coordinates of the vertex of the graph of $y = g(x)$.

   [1 mark]

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

   (c)

   Find the $x$-coordinates of the points where the graph of $y = g(x)$ meets the $x$-axis.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A company models its monthly profit, $P$ thousand dollars, from selling $x$ hundred items by $P(x) = 20 - (x - 6)^{2}$, for $0 \le x \le 12$. Next year a fixed extra cost of 4 thousand dollars per month is added, and nothing else changes. The new profit is $Q(x)$ thousand dollars.

   (a)

   Write down $Q(x)$ in terms of $P(x)$, and describe the transformation that maps the graph of $P$ onto the graph of $Q$. Hence find the maximum monthly profit next year and the number of items that gives it.

   [3 marks]

   (b)

   In a new market, the profit at every level of sales is 1.5 times the profit given by $Q$. The new profit is $R(x) = 1.5\,Q(x)$ thousand dollars. Describe the transformation that maps the graph of $Q$ onto the graph of $R$, and find the range of the number of items the company must sell in the new market so that it does not make a loss.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $f(x) = e^{x}$, $x \in \mathbb{R}$. The graph of $y = f(x)$ is transformed to the graph of $y = g(x)$ by the following transformations, in this order: a horizontal stretch with scale factor $\frac{1}{2}$; then a reflection in the $x$-axis; then a translation of 3 units in the positive $y$-direction.

   (a)

   (i) Find an expression for $g(x)$.
   (ii) Write down the equation of the horizontal asymptote of the graph of $g$, and the range of $g$.
   (iii) Find the exact $x$-coordinate of the point where the graph of $g$ meets the $x$-axis.

   [6 marks]

   (b)

   A student applies the same three transformations to the graph of $y = f(x)$ in the reverse order: first the translation, then the reflection, then the horizontal stretch. The result is the graph of $y = h(x)$.
   (i) Find an expression for $h(x)$.
   (ii) Show that the graphs of $y = g(x)$ and $y = h(x)$ never meet.
   (iii) Describe fully a single transformation that maps the graph of $y = h(x)$ onto the graph of $y = g(x)$.

   [6 marks]

   Total for question 4: 12 marks

End of questions