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So far, we have looked at solving and graphing quadratic polynomials. Recall that these are polynomials of degree two, x^{2}. We observed that such equations create a bellshaped curve. We also know that by the Fundamental Theorem of Algebra, quadratic polynomials have two roots. Now, what if we have a polynomial of degree three, x^{3 }? By the Fundamental Theorem of Algebra, such an equation would have at least three roots.
These are called cubic polynomials. In this case, what would the graph of a cubic look like compared to a quadratic? In this lesson, we shall look at graphing cubic polynomials. Let us first establish the definitions below.
A cubic polynomial is a polynomial of degree three. The standard form of a cubic polynomial is
where a, b, c and d are constants and a ≠ 0. A cubic graph is a graphical representation of a cubic polynomial.
Before we get into this topic, let us compare cubic graphs and quadratic graphs. We begin by noting the following definitions.
The axis of symmetry of a parabola (curve) is a vertical line that divides the parabola into two congruent (identical) halves.
The point of symmetry of a parabola is called the central point at which
We shall begin this topic by looking at the difference in properties of the cubic graph and the quadratic graph.
Recall that a quadratic expression is a polynomial of degree two and the standard form is ax^{2} + bx + c = 0, a ≠ 0.
Property  Quadratic Graph, y = x^{2}  Cubic Graph, y = x^{3} 
Basic Graph  The axis of symmetry is about the origin (0,0)  The point of symmetry is about the origin (0,0) 
Number of Roots(By Fundamental Theorem of Algebra)  2 solutions  3 solutions 
Domain  Set of all real numbers  Set of all real numbers 
Range  Set of all real numbers  Set of all real numbers 
Type of Function  Even  Odd 
Axis of Symmetry  Present  Absent 
Point of Symmetry  Absent  Present 
Turning Points  One: can either be a maximum or minimum value, depending on the coefficient of x^{2}  Zero: this indicates that the root has a multiplicity of three (the basic cubic graph has no turning points since the root x = 0 has a multiplicity of three, x^{3} = 0) 
OR  
Two: this indicates that the curve has exactly one minimum value and one maximum value 
There are three methods to consider when graphing cubic functions, namely
Transformation,
Factorization, and
Constructing a table of values.
We shall look at each technique in turn along with some worked examples to comprehensively demonstrate these methods.
In geometry, a transformation is a term used to describe a change in shape. Similarly, we can apply this concept in graph plotting. By modifying the coefficients or constants in a given cubic polynomial, we can vary the sketch of the curve. As before, the graph of the basic cubic function, y = x^{3}, is shown below.
Cubic polynomial graph, Aishah Amri  StudySmarter Originals
There are three ways in which we can transform this graph. This is illustrated in the table below.
Form of Cubic Polynomial  Change in Value  Plot of Graph 
y = ax^{3} 
 
Varying a changes the cubic function in the ydirection, that is, the coefficient a affects the steepness of the graph  
y = x^{3} + k 
 
Varying k shifts the cubic function up or down the yaxis by k units  
y = (x  h) ^{3} 
 
Varying h changes the cubic function along the xaxis by h units. 
Plot the graph of y = – 3x^{3} – 5.
Step 1: The coefficient of x^{3} is negative and has a factor of 3. Thus, we expect the basic cubic function to be inverted and steeper compared to the initial sketch.
Step 2: The term –5 indicates that the graph must move 5 units down the yaxis. Thus, taking our sketch from Step 1, we obtain the graph of y = – 3x^{3} – 5 as:
Plot the graph of y = (x + 4)^{3} + 2.
Step 1: The term (x + 4)^{3} indicates that the basic cubic graph shifts 4 units to the left of the xaxis.
Step 2: Finally, the term +2 tells us that the graph must move 2 units up the yaxis. Hence, taking our sketch from Step 1, we obtain the graph of y = (x + 4)^{3} + 2 as:
Through these transformations, we can generalize the change of coefficients a, k and h by the cubic polynomial
y = a(x – h)^{3} + k
This is known as the vertex form of cubic polynomials. Recall that this looks similar to the vertex form of quadratic polynomials. Notice that varying a, k and h follow the same concept in this case. The only difference here is that the power of (x – h) is 3 rather than 2!
For this technique of graphing cubic polynomials, we shall adopt the following steps:
Step 1: Factorize the given cubic polynomial
If the equation is in the form y = (x – a) (x – b) (x – c), we can proceed to the next step.
Step 2: Identify the xintercepts by setting y = 0
Step 3: Identify the yintercept by setting x = 0
Step 4: Plot the points and sketch the curve
In Step 1, we shall use factoring methods introduced in the previous topics.
Plot the graph of y = (x – 1) (x – 2) (x + 5).
Solution
Observe that the given function has been factorized completely. Thus, we can skip Step 1.
Step 2: Find the xintercepts
Setting y = 0, we obtain (x – 1) (x – 2) (x + 5) = 0.
Solving this, we obtain three roots, namely x = –5, x = 1 and x = 2.
Step 3: Find the yintercept
Plugging x = 0, we obtain y = (0 – 1) (0 – 2) (0 + 5) = (–1) (–2) (5) = 10
Thus, the yintercept is y = 10.
Step 4: Sketch the graph
As we have now identified the x and yintercepts, we can plot this on the graph and draw a curve to join these points together.
Notice that we obtain two turning points for this graph:
Plot the graph of y = (x + 1) (x^{2} – 6x + 9).
Solution
Step 1: Notice that the term x^{2} – 6x + 9 can be further factorized into a square of a binomial.
Using the formula for this form of Special Products, we obtain (x – 3)^{2}.
Thus, the given cubic polynomial becomes y = (x + 1) (x – 3)^{2}
Step 2: Setting y = 0, we obtain (x + 1) (x – 3)^{2} = 0.
Solving this, we have the single root x = –1 and the repeated root x = 3.
Note here that x = 3 has a multiplicity of 2.
Step 3: Plugging x = 0, we obtain y = (0 + 1) (0 – 3)^{2} = (1) (9) = 9
Thus, the yintercept is y = 9.
Step 4: Plotting these points and joining the curve, we obtain the following graph.
Again, we obtain two turning points for this graph:
For this case, since we have a repeated root at x = 3, the minimum value is known as an inflection point. Notice that from the left of x = 3, the graph is moving downwards, indicating a negative slope whilst from the right of x = 3, the graph is moving upwards, indicating a positive slope.
An inflection point is a point on the curve where it changes from sloping up to down or sloping down to up.
Plot the graph of y = x^{3} – 7x – 6, given that x = –1 is a solution to this cubic polynomial.
Solution
Step 1: By the Factor Theorem, if x = 1 is a solution to this equation, then (x + 1) must be a factor. Thus, we can rewrite the function as y = (x + 1) (ax^{2} + bx + c).
Note that in most cases, we may not be given any solutions to a given cubic polynomial. Hence, we need to conduct trial and error to find a value of x where the remainder is zero upon solving for y. Common values of x to try are 1, –1, 2, –2, 3 and –3.
To find the coefficients a, b and c in the quadratic equation ax^{2} + bx + c, we must conduct synthetic division as shown below.
By looking at the first three numbers in the last row, we obtain the coefficients of the quadratic equation and thus, our given cubic polynomial becomes y = (x + 1) (x^{2} – x – 6).
We can further factorize the expression x^{2} – x – 6 as (x – 3) (x + 2).
Thus, the complete factorized form of this function is y = (x + 1) (x – 3) (x + 2)
Step 2: Setting y = 0, we obtain (x + 1) (x – 3) (x + 2) = 0,
Solving this, we obtain three roots: x = –2, x = –1 and x = 3.
Step 3: Plugging x = 0, we obtain y = (0 + 1) (0 – 3) (0 + 2) = (1) (–3) (2) = –6
Thus, the yintercept is y = –6.
Step 4: The graph for this given cubic polynomial is sketched below.
Once more, we obtain two turning points for this graph:
Plot the graph of y =  (2x – 1) (x^{2} – 1).
Solution
Firstly, notice that there is a negative sign before the equation above. This means that the graph will take the shape of an inverted (standard) cubic polynomial graph. In other words, this curve will first open up and then open down.
Step 1: We first notice that the binomial (x^{2} – 1) is an example of a perfect square binomial. By the principle of Special Products, we can further factorize this expression as (x + 1) (x – 1). Thus, the complete factored form of this equation is y = – (2x – 1)(x + 1) (x – 1).
Step 2: Setting y = 0, we obtain
Solving this, we obtain three roots: x = –1, x = and x = 1
Step 3: Plugging x = 0, we obtain
Thus, the yintercept is y = –1.
Step 4: The graph for this given cubic polynomial is sketched below. Be careful and remember the negative sign in our initial equation! The cubic graph will is flipped here.
In this case, we obtain two turning points for this graph:
Before we begin this method of graphing, we shall introduce The Location Principle.
The Location Principle
Suppose y = f(x) represents a polynomial function.
Let a and b be two numbers in the domain of f such that f(a) < 0 and f(b) > 0.
Then the function has at least one real zero between a and b.
The Location Principle, Glencoe McGrawHill, Algebra 2 (2008)
The Location Principle will help us determine the roots of a given cubic function since we are not explicitly factorizing the expression. For this technique, we shall make use of the following steps.
Step 1: Evaluate f(x) for a domain of x values and construct a table of values (we will only consider integer values)
Step 2: Locate the zeros of the function
Step 3: Identify the maximum and minimum points
Step 4: Plot the points and sketch the curve
This method of graphing can be somewhat tedious as we need to evaluate the function for several values of x. However, this technique may be helpful in estimating the behaviour of the graph at certain intervals.
Note that in this method, there is no need for us to completely solve the cubic polynomial. We are simply graphing the expression using the table of values constructed. The trick here is to calculate several points from a given cubic function and plot it on a graph which we will then connect together to form a smooth, continuous curve.
Graph the cubic function f(x) = x^{3} – 3x^{2} + 5.
Step 1: Let us evaluate this function between the domain x = –2 and x = 4. Constructing the table of values, we obtain the following range of values for f(x).
x  f(x) 
–2  –15 
–1  1 
0  5 
1  3 
2  1 
3  5 
4  21 
Step 2: Notice that between x = –2 and x = –1 the value of f(x) changes sign. The Location Principle indicates that there is a zero in between these xvalues.
Step 3: We first observe the interval between x = –1 and x = 1. The value of f(x) at x = 0 seems to be greater compared to its neighbouring points. This indicates that we have a relative maximum.
Similarly, notice that the interval between x = 1 and x = 3 contains a relative minimum since the value of f(x) at x = 2 is lesser than its surrounding points.
We use the term relative maximum or minimum here as we are only guessing the location of the maximum or minimum point given our table of values.
Step 4: Now that we have these values and we have concluded the behaviour of the function between this domain of x, we can sketch the graph as shown below.
Form of Cubic Polynomial  Description  Change in Value 
y = ax^{3}  Varying a changes the cubic function in the ydirection 

y = x^{3} + k  Varying k shifts the cubic function up or down the yaxis by k units 

y = (x  h) ^{3}  Varying h changes the cubic function along the xaxis by h units 

To graph cubic polynomials, we must identify the vertex, reflection, yintercept and xintercepts
To find the expression of a cubic polynomial from a graph, we must identify the zeros (xintercepts)
To draw the graph of a cubic polynomial, we must identify the vertex, reflection, yintercept and xintercepts
To find the formula of a cubic polynomial from a graph, we must identify the zeros (xintercepts)
A cubic graph is a graph with three vertices (vertices of degree 3)
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