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Cubic Polynomial Graphs

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Cubic Polynomial Graphs

So far, we have looked at solving and graphing quadratic polynomials. Recall that these are polynomials of degree two, x2. We observed that such equations create a bell-shaped 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, x3 ? 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

  1. the curve divides into two equal parts (that are of equal distance from the central point)
  2. both parts face different directions

Cubic Graphs Vs. Quadratic Graphs

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 ax2 + bx + c = 0, a ≠ 0.

Property
Quadratic Graph, y = x2
Cubic Graph, y = x3
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 x2

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, x3 = 0)

OR

Two: this indicates that the curve has exactly one minimum value and one maximum value

Sketching Cubic Functions

There are three methods to consider when graphing cubic functions, namely

  1. Transformation,

  2. Factorization, and

  3. Constructing a table of values.

We shall look at each technique in turn along with some worked examples to comprehensively demonstrate these methods.

Transformation

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 = x3, 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 = ax3

  • If a is large (> 1), the graph becomes steeper (red line)
  • If a is small (0 < a < 1), the graph becomes flatter (blue line)
  • If a is negative, the graph becomes inverted (green line)

Varying a changes the cubic function in the y-direction, that is, the coefficient a affects the steepness of the graph

y = x3 + k

  • If k is negative, the graph moves down k units in the y-axis (blue line)
  • If k is positive, the graph moves up k units in the y-axis (green line)

Varying k shifts the cubic function up or down the y-axis by k units

y = (x - h) 3

  • If h is negative, the graph shifts h units to the left of the x-axis (blue line)
  • If h is positive, the graph shifts h units to the right of the x-axis (green line)

Varying h changes the cubic function along the x-axis by h units.

Plot the graph of y = – 3x3 – 5.

Step 1: The coefficient of x3 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 y-axis. Thus, taking our sketch from Step 1, we obtain the graph of y = – 3x3 – 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 x-axis.

Step 2: Finally, the term +2 tells us that the graph must move 2 units up the y-axis. 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!

Factorization

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 x-intercepts by setting y = 0

Step 3: Identify the y-intercept 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 x-intercepts

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 y-intercept

Plugging x = 0, we obtain y = (0 – 1) (0 – 2) (0 + 5) = (–1) (–2) (5) = 10

Thus, the y-intercept is y = 10.

Step 4: Sketch the graph

As we have now identified the x and y-intercepts, 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:

  1. a maximum value between the roots x = –5 and x = 1
  2. a minimum value between the roots x = 1 and x = 2.

Plot the graph of y = (x + 1) (x2 – 6x + 9).

Solution

Step 1: Notice that the term x2 – 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 y-intercept 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:

  1. a maximum value between the roots x = –1 and x = 3
  2. a minimum value at x = 3.

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 = x3 – 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) (ax2 + 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 ax2 + 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) (x2 – x – 6).

We can further factorize the expression x2 – 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 y-intercept 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:

  1. a maximum value between the roots x = –2 and x = –1
  2. a minimum value between the roots x = –1 and x = 3.

Plot the graph of y = - (2x – 1) (x2 – 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 (x2 – 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 y-intercept 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:

  1. a minimum value between the roots x = –1 and x =
  2. a maximum value between the roots x = and x = 1

Constructing a Table of Values

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 McGraw-Hill, 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) = x3 – 3x2 + 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 x-values.

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.

Cubic Polynomial Graphs - Key takeaways

  • A cubic graph has three roots and two turning points
  • Sketching by the transformation of cubic graphs
    Form of Cubic Polynomial
    Description
    Change in Value

    y = ax3

    Varying a changes the cubic function in the y-direction
    • If a is large (> 1), the graph becomes steeper
    • If a is small (0 < a < 1), the graph becomes flatter
    • If a is negative, the graph becomes inverted

    y = x3 + k

    Varying k shifts the cubic function up or down the y-axis by k units
    • If k is negative, the graph moves down k units
    • If k is positive, the graph moves up k units

    y = (x - h) 3

    Varying h changes the cubic function along the x-axis by h units
    • If h is negative, the graph shifts h units to the left
    • If h is positive, the graph shifts h units to the right
  • Graphing by factorization of cubic polynomials
    1. Factorize the given cubic polynomial
    2. Identify the x-intercepts by setting y = 0
    3. Identify the y-intercept by setting x = 0
    4. Plot the points and sketch the curve
  • Plotting by constructing a table of values
    1. Evaluate f(x) for a domain of x values and construct a table of values
    2. Locate the zeros of the function
    3. Identify the maximum and minimum points
    4. Plot the points and sketch the curve

Frequently Asked Questions about Cubic Polynomial Graphs

To graph cubic polynomials, we must identify the vertex, reflection, y-intercept and x-intercepts 

To find the expression of a cubic polynomial from a graph, we must identify the zeros (x-intercepts)

To draw the graph of a cubic polynomial, we must identify the vertex, reflection, y-intercept and x-intercepts 

To find the formula of a cubic polynomial from a graph, we must identify the zeros (x-intercepts)

A cubic graph is a graph with three vertices (vertices of degree 3)

Final Cubic Polynomial Graphs Quiz

Question

What is a cubic function? State the general form of a cubic equation. 

Show answer

Answer

A cubic function is a polynomial of degree three. The general form of a cubic equation is ax3 + bx2 + cx + d = 0 where a  0.

Show question

Question

Do cubic functions have axes of symmetry? 

Show answer

Answer

No 

Show question

Question

What are the first two steps when plotting cubic functions? 

Show answer

Answer

Find the x and y intercepts 

Show question

Question

For the cubic function y = ax3, what does varying the coefficient a do to the graph? 

Show answer

Answer

Varying the coefficient a changes the y-direction of the standard cubic graph

Show question

Question

For the cubic function y = x3 + k, what does varying the coefficient k do to the graph? 


Show answer

Answer

Varying the coefficient k moves the cubic graph up or down the y-axis by k units. 

Show question

Question

For the cubic function y = (x - h)3, what does varying the coefficient h do to the graph? 



Show answer

Answer

Varying the h moves the basic cubic along the x-axis by h units

Show question

Question

For the cubic function y = ax3, when a is large, what happens to the cubic graph? 

Show answer

Answer

The cubic graph becomes steeper

Show question

Question

For the cubic function y = ax3, when a is small, what happens to the cubic graph? 


Show answer

Answer

The cubic graph becomes flatter

Show question

Question

For the cubic function y = ax3, when a is negative, what happens to the cubic graph? 


Show answer

Answer

The cubic graph becomes inverted

Show question

Question

For the cubic function y = x+ k, when k is positive, what happens to the cubic graph? 


Show answer

Answer

The cubic graph moves up the y-axis

Show question

Question

For the cubic function y = x+ k, when k is negative, what happens to the cubic graph? 

Show answer

Answer

The cubic graph moves down the y-axis

Show question

Question

For the cubic function y = (x - h)3, when h is positive, what happens to the cubic graph? 

Show answer

Answer

The cubic graph moves to the right of the x-axis

Show question

Question

For the cubic function y = (x - h)3, when h is negative, what happens to the cubic graph? 

Show answer

Answer

The cubic graph moves to the left of the x-axis

Show question

Question

How many turning points do cubic graphs have? What are the properties of these turning points?

Show answer

Answer

Cubic graphs have two turning points. A maximum point and a minimum point.

Show question

Question

What are the three steps in plotting cubic functions? 

Show answer

Answer

  1. Find the x-intercepts
  2. Locate the y-intercepts
  3. Create a table of values
  4. Identify the minimum and maximum points
  5. Sketch the graph 

Show question

Question

What is the Location Principle in the context of a cubic equation?

Show answer

Answer

Suppose y = f(x) is a cubic polynomial function. If a and b are two values such that f(a) < 0 and f(b) > 0, then the function has at least one real zero between a and b.

Show question

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