Sketching polynomial and reciprocal graphsAQA A-Level Maths: Subtopic test

1. 1
   The curve $C$ has equation $y=(x+2)(x-1)^2$.

   (a)

   Which statement about where $C$ meets the $x$-axis is correct?

   [1 mark]

   • A$C$ crosses the $x$-axis at $x=-2$ and at $x=1$
   • B$C$ crosses the $x$-axis at $x=-2$ and touches it at $x=1$
   • C$C$ touches the $x$-axis at $x=-2$ and crosses it at $x=1$
   • D$C$ crosses the $x$-axis at $x=2$ and touches it at $x=-1$

   (b)

   Find the $y$-intercept of $C$.

   [1 mark]

   • A$-2$
   • B$1$
   • C$0$
   • D$2$

   (c)

   Write down the coordinates of the point where $C$ touches the $x$-axis, and describe the behaviour of $y$ as $x\to-\infty$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $D$ has equation $y=\dfrac{4}{x^2}$.

   (a)

   In which quadrants does $D$ lie?

   [1 mark]

   • AFirst and second
   • BFirst and third
   • CThird and fourth
   • DAll four

   (b)

   The point $(-2,k)$ lies on $D$. Find $k$.

   [1 mark]

   • A$-1$
   • B$-2$
   • C$1$
   • D$2$

   (c)

   Write down the equations of the asymptotes of $D$, and explain why $D$ never meets the $x$-axis.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $E$ has equation $y=x^2(3-x)$.

   (a)

   Find the coordinates of the points where $E$ meets the $x$-axis and state, with a reason, whether $E$ crosses or touches the $x$-axis at each of them.

   [3 marks]

   (b)

   Find the coordinates of the turning points of $E$ and determine their nature.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curves $C_1$ and $C_2$ have equations $C_1: y=(x-1)(x+3)(x-4)$ and $C_2: y=\dfrac{12}{x}$.

   (a)

   Find the coordinates of the points where $C_1$ meets the axes, and describe the behaviour of $C_1$ as $x\to\infty$ and as $x\to-\infty$.

   [6 marks]

   (b)

   Explain why $C_1$ and $C_2$ have no point of intersection with $1<x<4$ or with $-3<x<0$, and show that they meet at least once for $x>4$ and at least once for $x<-3$.

   [6 marks]

   Total for question 4: 12 marks

End of questions