GRAPHS OF FUNCTIONS IT LESSON 1 – CUBIC FUNCTIONS
(SEC 3 ELEMENTARY MATHEMATICS)

Software: LiveMath (previously known as MathView)
Thinking Skills: Induction and Deduction
At the end of the lesson, the students should be able to:
(1) sketch the graphs of cubic functions,
(2) describe the characteristics of the graphs of cubic functions.

A.  GRAPHS OF CUBIC FUNCTIONS 1

If you cannot view the LiveMath documents below, click here to troubleshoot.
 

1. The first document (IT3EMGraphFnCubic1.the) shows the graph of the cubic function y=ax3+bx2+cx+d where a is not equal to 0. Change the value of a to 1, the value of b to 0, the value of c to –3 and the value of d to 2 in order to get the curve y=x3–3x+2.   2. Does the curve have any turning points? If yes, record their coordinates in the table in the worksheet.  Hint: You can use the knife button (first button on top right hand corner of graph) to draw a region enclosing the turning point and then release. This will zoom in on the selected region. Repeat the process until you can read the coordinates. If you mess up the graph, click Reload or Refresh.   3. Does the curve have any line of symmetry? If yes, find and record their equations in the same table.   4. Does the curve have any rotational symmetry? If yes, find and record the coordinates of the centre of rotational symmetry and its order in the same table.   5. Make a sketch of the curve in the same table, showing clearly the turning points (if any). Do NOT draw the axes.   6. Repeat Steps 1 to 5 for the following curves.  (a) y = 3x2–x3    (b) y = x3+2x–1    (c) y = –2x3

Question 1:	Infer by induction the number of turning points a cubic curve can have. _________  or  _________ .
	Do they help to determine the shape of a cubic curve? *Yes/No. (*delete accordingly)
Question 2:	Infer by induction whether a cubic curve has any line of symmetry.  __________________________________________________
Question 3:	Infer by induction the order of rotational symmetry a cubic curve has.  ____________________________________________________
Question 4:	If a cubic curve has turning points, how do they help in locating the centre of rotational symmetry?  ______________________________________________________________

B.  GRAPHS OF CUBIC FUNCTIONS 2
 

1. The 2nd document (IT3EMGraphFnCubic2.the) shows the animation of the graphs of the cubic functions y=x3–2x+d where d increases from –3 to 4 in steps of 1. Click on the graph to start the animation.     Note: The animation will stop at 3. It will not include the last value.     Question  5: With reference to the curve y=x3–2x, what effect does the constant d have?  If d>0, the reference curve is shifted ______. If d<0, the reference curve is shifted ______.     2. The second animation shows how the coefficient of x3 affects the graphs of the cubic functions y=ax3–3x where a increases from –3 to 4 in steps of 1. Click on the graph to start the animation.     Question  6: How does the coefficient of x3 affect the shapes of the graphs of cubic functions?  If a>0, when x becomes very big, the curve goes *up/down. (*delete accordingly)  [In mathematical terms, we say that as x tends to positive infinity, y tends to positive infinity.]  If a<0, when x becomes very big, the curve goes ________ .  [In mathematical terms, we say that as x tends to positive infinity, y tends to negative infinity.]

C.  GRAPHS OF CUBIC FUNCTIONS 3 (ENRICHMENT)
 

1. The 3rd document (IT3EMGraphFnCubic3.the) shows the animation of the graphs of the cubic functions y=x3+bx2–3x+1 where b increases from –2 to 3 in steps of 1. Click on the graph to start the animation.     Question 7: Where does the centre of rotation of the graph of y=x3–3x+1 lie?  When b=0, the centre of rotation lies on the _____ - axis.  Hint: Stop the animation when b=0 and observe where is the centre of rotation.   Question 8: With reference to the graph of y=x3–3x+1, what effect does the coefficient of x2 have?  If b is not equal to 0, the centre of rotation is shifted away from the _____ - axis.      2. The second animation shows how the coefficient of x affects the graphs of the cubic functions y=x3+cx where c increases from –3 to 4 in steps of 1. Click on the graph to start the animation.     Question 9: How does the coefficient of x affect the shapes of the graphs of cubic functions?  It may change the number of turning points from ______ to _____ . To understand the following, please note that a stationary point is a point on the curve where the gradient is zero. There are 3 types of stationary points: maximum turning point, minimum turning point and point of inflexion.  Stop the animation when c=0 and observe that the origin is a point of inflexion (where the gradient is also zero). But when c>0, there is no stationary point at all - the curve is an increasing function as the gradient is always positive.  This is a summary of the effects of the following terms:    Term in x2: the  only term that shifts the centre of rotation away from the y-axis; create 2 turning points (e.g. y=x3–x2) but this effect may be cancelled off by the term in x (e.g. y=x3+3x2+3x has no turning points)    Term in x: may or may not create 2 turning points (e.g. y=x3–x has 2 turning points but y=x3+x has no turning points)

D.  EXERCISE
 

1.	Do NOT use your computer to do the exercise. You can check your answers after that.
2.	Apply what you have learnt to deduce the shapes of the following curves and sketch them, showing clearly the turning points (if any). Do NOT draw the axes.
	(a)    y = x3        (b)    y = –x3        (c)    y = x3–3x2+2x        (d)    y = –2x3+4x2–1