Rational and reciprocal functionsIB MYP Maths Extended: Subtopic test

1. 1
   Consider the function $f(x)=\dfrac{6}{x}$, for $x\neq0$.

   (a)

   Which pair gives the equations of the asymptotes of the graph of $y=f(x)$?

   [1 mark]

   • A$x=6$ and $y=6$
   • B$x=0$ only
   • C$x=0$ and $y=0$
   • D$y=6$ and $x=0$

   (b)

   Which point lies on the graph of $y=f(x)$?

   [1 mark]

   • A$(-3,2)$
   • B$(-3,-2)$
   • C$(2,12)$
   • D$(0,6)$

   (c)

   Explain why the graph of $y=f(x)$ crosses neither axis.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the function $g(x)=\dfrac{2}{x-3}+1$, for $x\neq3$.

   (a)

   Which pair gives the equations of the asymptotes of the graph of $y=g(x)$?

   [1 mark]

   • A$x=3$ and $y=1$
   • B$x=-3$ and $y=1$
   • C$x=3$ and $y=2$
   • D$x=1$ and $y=3$

   (b)

   Where does the graph of $y=g(x)$ cross the $y$-axis?

   [1 mark]

   • A$\left(0,-\frac23\right)$
   • B$(0,1)$
   • C$(0,-3)$
   • D$\left(0,\frac13\right)$

   (c)

   Find the exact coordinates of the point where the graph of $y=g(x)$ crosses the $x$-axis.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the function $h(x)=\dfrac{x+1}{x-2}$, for $x\neq2$.

   (a)

   Show that $h(x)=1+\dfrac{3}{x-2}$, and hence write down the equations of the asymptotes of the graph of $y=h(x)$.

   [3 marks]

   (b)

   Find the coordinates of the points where the graph of $y=h(x)$ crosses the axes. State whether $h(x)$ is positive or negative for $-1<x<2$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A school hires a coach with $50$ seats for a trip. The hire costs $1200$ dirhams, shared equally between the $n$ students who go, and each student also pays an entry fee of $15$ dirhams. The cost per student, $c$ dirhams, is modelled by $c=\dfrac{1200}{n}+15$, for $10\leq n\leq50$.

   (a)

   (i) Find the cost per student when $24$ students go.
   (ii) State the equation of the horizontal asymptote of the graph of $c$ against $n$, and explain what it means in this context.
   (iii) Find the value of $n$ for which the cost per student is $39$ dirhams.

   [6 marks]

   (b)

   (i) The school will not charge more than $40$ dirhams per student. Find the smallest number of students that makes this possible.
   (ii) A teacher says: ‘If $100$ students go on two coaches, the model shows each pays only $27$ dirhams.’ Evaluate this claim.

   [6 marks]

   Total for question 4: 12 marks

End of questions