Cubic and polynomial graphsIB MYP Maths Extended: Subtopic test

1. 1
   Consider the cubic function $f(x)=(x+2)(x-1)(x-3)$.

   (a)

   Which values of $x$ give the $x$-intercepts of the graph of $y=f(x)$?

   [1 mark]

   • A$x=2,\ -1,\ -3$
   • B$x=-2,\ 1,\ -3$
   • C$x=-2,\ -1,\ 3$
   • D$x=-2,\ 1,\ 3$

   (b)

   What is the $y$-intercept of the graph of $y=f(x)$?

   [1 mark]

   • A$6$
   • B$0$
   • C$-6$
   • D$2$

   (c)

   Write down the values of $x$ for which $f(x)>0$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the cubic function $g(x)=x^3-4x$.

   (a)

   Find all solutions of $x^3-4x=0$.

   [1 mark]

   • A$x=0$ and $x=2$
   • B$x=-4,\ 0,\ 4$
   • C$x=-2,\ 0,\ 2$
   • D$x=2$ only

   (b)

   How many turning points does the graph of $y=g(x)$ have?

   [1 mark]

   • A0
   • B2
   • C1
   • D3

   (c)

   Use technology to find the coordinates of the local minimum point of the graph of $y=g(x)$. Give each coordinate to 3 significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Sam investigates the family of cubic graphs $y=x^3-kx$ for different positive values of $k$. Using graphing software, Sam finds that for $k=1$ the graph crosses the $x$-axis at $-1$, $0$ and $1$; for $k=4$ it crosses at $-2$, $0$ and $2$; and for $k=9$ it crosses at $-3$, $0$ and $3$.

   (a)

   Describe the pattern in the $x$-intercepts of $y=x^3-kx$ and write a general rule for them in terms of $k$. Verify your rule for $k=16$.

   [3 marks]

   (b)

   Use technology to find the coordinates of the turning points when $k=3$ and when $k=12$. Hence suggest a rule for the $x$-coordinates of the turning points in terms of $k$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A packaging company makes open boxes from square pieces of card of side 30 cm. A square of side $x$ cm is cut from each corner and the sides are folded up. The volume of the box is $V=x(30-2x)^2$ cm$^3$, where $0<x<15$.

   (a)

   (i) Show that $V=4x^3-120x^2+900x$.
   (ii) Write down the values of $x$ for which $V=0$ and explain why only $0<x<15$ gives a possible box.
   (iii) Use technology to find the value of $x$ that gives the largest volume, and state this volume.

   [6 marks]

   (b)

   The company wants every box to hold at least $1800$ cm$^3$. Use technology to find the range of values of $x$ that meet this requirement, and justify whether cutting squares of side $8$ cm is suitable.

   [6 marks]

   Total for question 4: 12 marks

End of questions