1. 1
   The function $f$ is defined by $f(x) = 2x^2 - 12x + 10$, for $x \in \mathbb{R}$.

   (a)

   Find the equation of the axis of symmetry of the graph of $y = f(x)$.

   [1 mark]

   • A$x = 3$
   • B$x = -3$
   • C$x = 6$
   • D$x = 5$

   (b)

   Which expression is $f(x)$ written in the form $a(x - h)^2 + k$?

   [1 mark]

   • A$2(x - 3)^2 + 1$
   • B$2(x + 3)^2 - 8$
   • C$2(x - 3)^2 + 28$
   • D$2(x - 3)^2 - 8$

   (c)

   Write $f(x)$ in the form $2(x - p)(x - q)$, and hence write down the $x$-intercepts of the graph of $y = f(x)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $g$ is defined by $g(x) = -3(x + 2)^2 + 12$, for $x \in \mathbb{R}$.

   (a)

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

   [1 mark]

   • A$(2, 12)$
   • B$(-2, 12)$
   • C$(-2, -12)$
   • D$(2, -12)$

   (b)

   Find the $y$-intercept of the graph of $y = g(x)$.

   [1 mark]

   • A$(0, 24)$
   • B$(0, 12)$
   • C$(0, 0)$
   • D$(0, 6)$

   (c)

   Write $g(x)$ in the form $-3(x - p)(x - q)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A footbridge is supported by an arch in the shape of a parabola. The height of the arch above the ground is $h(x)$ metres at a horizontal distance of $x$ metres from one foot of the arch. The arch meets the ground at $x = 0$ and at $x = 40$, and its maximum height is 12 m.

   (a)

   Show that $h(x) = -0.03x(x - 40)$.

   [3 marks]

   (b)

   A vehicle 8 m wide and 11 m high drives along the ground under the arch, keeping its centre directly below the highest point of the arch. Determine whether the vehicle can pass under the arch. Justify your answer.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The quadratic function $f$ can be written in the form $f(x) = a(x + 1)^2 + 18$. The graph of $y = f(x)$ passes through the point $(2, 0)$. A second quadratic function $g$ has the same zeros as $f$, and the graph of $y = g(x)$ passes through the point $(0, 24)$.

   (a)

   (i) Find the value of $a$.
   (ii) Show that $f(x) = -2x^2 - 4x + 16$.
   (iii) Write $f(x)$ in the form $a(x - p)(x - q)$.

   [6 marks]

   (b)

   (i) Find $g(x)$ in the form $b(x - p)(x - q)$.
   (ii) Show that $g(x) = 1.5f(x)$ for all values of $x$.
   (iii) Write down the coordinates of the vertex of the graph of $y = g(x)$, and explain why the graphs of $f$ and $g$ have the same axis of symmetry.

   [6 marks]

   Total for question 4: 12 marks

End of questions