The modulus functionAQA A-Level Maths: Subtopic test

1. 1
   The graph of $y=|2x-6|$ is considered.

   (a)

   State the coordinates of the vertex of the graph.

   [1 mark]

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

   (b)

   State the $y$-coordinate of the point where the graph crosses the $y$-axis.

   [1 mark]

   • A$-6$
   • B$6$
   • C$3$
   • D$0$

   (c)

   Write $y=|2x-6|$ without modulus signs, stating the values of $x$ for which each form applies.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $y=|x+4|$ and the horizontal line $y=k$, where $k$ is a constant, are considered.

   (a)

   For which values of $k$ does the equation $|x+4|=k$ have exactly two solutions?

   [1 mark]

   • A$k>0$
   • B$k=0$
   • C$k<0$
   • Dall real $k$

   (b)

   Which statement correctly describes how the graph of $y=|x+4|$ is obtained from the line $y=x+4$?

   [1 mark]

   • AThe whole line is reflected in the $y$-axis.
   • BThe part of the line to the left of the $y$-axis is reflected in the $y$-axis.
   • CThe line is translated $4$ units up.
   • DThe part of the line below the $x$-axis is reflected in the $x$-axis.

   (c)

   Find the coordinates of the points where the curve $y=|x+4|$ meets the axes.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=|4-3x|$.

   (a)

   Find the coordinates of the vertex of $C$ and of the point where $C$ meets the $y$-axis, and state the equation of the line of symmetry of $C$.

   [3 marks]

   (b)

   The line $y=2$ meets $C$ at two points. Find their $x$-coordinates and the distance between the points.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=|2x-5|$ and the line $L$ has equation $y=x+1$.

   (a)

   Find the coordinates of the vertex of $C$ and of the point where $C$ meets the $y$-axis, and find the $x$-coordinates of the points where $L$ meets $C$.

   [6 marks]

   (b)

   The line $y=x+k$ is drawn for different values of $k$. Use the shape of $C$ to find, with justification, the values of $k$ for which the line meets $C$ (i) exactly once, (ii) exactly twice, (iii) not at all.

   [6 marks]

   Total for question 4: 12 marks

End of questions