Quadratic functions and graphsAQA A-Level Maths: Subtopic test

1. 1
   The quadratic function $f$ is defined by $f(x)=x^{2}-6x+13$.

   (a)

   Which of these is $f(x)$ written in completed square form?

   [1 mark]

   • A$(x-3)^2+13$
   • B$(x+3)^2+4$
   • C$(x-3)^2-4$
   • D$(x-3)^2+4$

   (b)

   Find the coordinates of the minimum point of the graph of $y=f(x)$.

   [1 mark]

   • A$(-3,4)$
   • B$(3,13)$
   • C$(3,4)$
   • D$(3,-4)$

   (c)

   Show that the equation $f(x)=0$ has no real roots.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The quadratic function $g$ is defined by $g(x)=2x^{2}+8x-3$.

   (a)

   Which of these is $g(x)$ in the form $a(x+p)^2+q$?

   [1 mark]

   • A$2(x+2)^2-3$
   • B$2(x+2)^2-11$
   • C$2(x+4)^2-35$
   • D$2(x+2)^2+5$

   (b)

   Find the equation of the line of symmetry of the graph of $y=g(x)$.

   [1 mark]

   • A$x=-2$
   • B$x=2$
   • C$x=-4$
   • D$x=-11$

   (c)

   Solve $g(x)=0$, giving your answers in exact form.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The equation $x^{2}-kx+(k+3)=0$, where $k$ is a constant.

   (a)

   Find the values of $k$ for which the equation has a repeated root.

   [3 marks]

   (b)

   Find the set of values of $k$ for which the equation has two distinct real roots.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A farmer uses 40 m of fencing to make three sides of a rectangular pen. The fourth side is an existing straight wall, so it needs no fencing. The two sides perpendicular to the wall each have length $x$ metres, and the area of the pen is $A$ m$^2$.

   (a)

   (i) Show that $A=40x-2x^2$.
   (ii) By completing the square, find the maximum value of $A$ and the value of $x$ at which it occurs.
   (iii) State the set of possible values of $x$.

   [6 marks]

   (b)

   (i) Using the discriminant, show that the area of the pen cannot be 210 m$^2$.
   (ii) Find the set of values of $x$ for which the area is at least 150 m$^2$.

   [6 marks]

   Total for question 4: 12 marks

End of questions