Graphs of Cubic Polynomials, Curve Sketching and Solutions to Simple Cubic Equations

Cubic Polynomials and Equations

A cubic polynomial is a polynomial of degree 3.

where a is nonzero.

An equation involving a cubic polynomial is called a cubic equation and is of the form f(x) = 0. There is also a closed-form solution known as the cubic formula which exists for the solutions of an arbitrary cubic equation. A cubic polynomial is represented by a function of the form. And f(x) = 0 is a cubic equation. The points at which this curve cuts the X-axis are the roots of the equation.

Here is a tutorial on Quadratic Equations, Cubic and Higher Order Equations - Using Factorization and Formula- A tutorial with solved problems and a Quiz

Examples of Cubic Curves And Equations

Example 1

Graph of y = f(x) = (x+8)(x+10)(x+20) = x³ + 38x² + 440x + 1600

A few computed points on the curve, apart from the zero(s) which are known:
(-21,-143), (-18,160), (-15,175), (-12,64), (-9,-11), (-6,112), (-3,595)

Characteristics of the Graph Plot, Curve Sketching of Cubic Curves

If the given cubic function is: f(x) = ax³+ bx²+ cx + d
The derivative of this function is: f'(x) = 3ax² + 2bx + c

The function given to us us f(x) = (x+8)(x+10)(x+20) = x³ + 38x² + 440x + 1600

And the derivative for this is f'(x) = 3x² + 76x + 1

Consider the cubic equation f(x) = (x+8)(x+10)(x+20) = x³ + 38x² + 440x + 1600 = 0

The roots of this cubic equation are at:

(x - (-8)) = 0 => x = -8,

OR (x - (-10)) = 0 => x = -10,

OR (x - (-20)) = 0 => x =-20

This cubic equation has real and unique roots at -8, -10, -20.
The plotted curve cuts the x-axis at these values of x: i.e, these are the zeroes of the given cubic polynomial

The turning or stationary points is where f'(x) = 0 => 3x² + 76x + 1 = 0 => x = -16.38, x = -8.95

These are also called the "critical" points where the derivative is zero.

Coming to other geometrical features of this curve: The y-intercept of this curve is at y=1600.
And the second derivative of this curve becomes zero at x = -12.67. At this point the curve changes concavity. A cubic curve has point symmetry around the point of inflection or inflexion.

The zeroes of a polynomial, if they are known, and the coefficients of that polynomial are two different sets of numbers that have interesting relations. If we know the zeroes, then we can write down algebraic expressions for the coefficients. Going the other way is much harder and cannot be done in general.

A cubic function has a bit more variety in its shape than the quadratic polynomials which are always parabolas. We can get a lot of information from the factorization of a cubic function. We get a fairly generic cubic shape when we have three distinct linear factors.

Example 2

Graph of y = f(x) = (x+14)(x- 0)(x+20) = x³ + 34x² + 280x + 0

A few computed points on the curve, apart from the zero(s) which are known:
(-21,-147), (-18,144), (-15,75), (-12,-192), (-9,-495), (-6,-672), (-3,-561), (0,0), (3,1173)

Characteristics of the Graph Plot, Curve Sketching of Cubic Curves

If the given cubic function is: f(x) = ax³+ bx²+ cx + d
The derivative of this function is: f'(x) = 3ax² + 2bx + c

The function given to us us f(x) = (x+14)(x- 0)(x+20) = x³ + 34x² + 280x + 0

And the derivative for this is f'(x) = 3x² + 68x + 1

Consider the cubic equation f(x) = (x+14)(x- 0)(x+20) = x³ + 34x² + 280x + 0 = 0

The roots of this cubic equation are at:

(x - (-14)) = 0 => x = -14,

OR (x - (0)) = 0 => x = 0,

OR (x - (-20)) = 0 => x =-20

This cubic equation has real and unique roots at -14, 0, -20.
The plotted curve cuts the x-axis at these values of x: i.e, these are the zeroes of the given cubic polynomial

The turning or stationary points is where f'(x) = 0 => 3x² + 68x + 1 = 0 => x = -17.26, x = -5.41

These are also called the "critical" points where the derivative is zero.

Coming to other geometrical features of this curve: The y-intercept of this curve is at y=0.
And the second derivative of this curve becomes zero at x = -11.33. At this point the curve changes concavity. A cubic curve has point symmetry around the point of inflection or inflexion.

The zeroes of a polynomial, if they are known, and the coefficients of that polynomial are two different sets of numbers that have interesting relations. If we know the zeroes, then we can write down algebraic expressions for the coefficients. Going the other way is much harder and cannot be done in general.

Example 3

Graph of y = f(x) = (x- 4)(x+10)(x+20) = x³ + 26x² + 80x -800

A few computed points on the curve, apart from the zero(s) which are known:
(-21,-275), (-18,352), (-15,475), (-12,256), (-9,-143), (-6,-560), (-3,-833), (0,-800), (3,-299), (6,832), (9,2755)

Characteristics of the Graph Plot, Curve Sketching of Cubic Curves

If the given cubic function is: f(x) = ax³+ bx²+ cx + d
The derivative of this function is: f'(x) = 3ax² + 2bx + c

The function given to us us f(x) = (x- 4)(x+10)(x+20) = x³ + 26x² + 80x -800

And the derivative for this is f'(x) = 3x² + 52x + 1

Consider the cubic equation f(x) = (x- 4)(x+10)(x+20) = x³ + 26x² + 80x -800 = 0

The roots of this cubic equation are at:

(x - (4)) = 0 => x = 4,

OR (x - (-10)) = 0 => x = -10,

OR (x - (-20)) = 0 => x =-20

This cubic equation has real and unique roots at 4, -10, -20.
The plotted curve cuts the x-axis at these values of x: i.e, these are the zeroes of the given cubic polynomial

The turning or stationary points is where f'(x) = 0 => 3x² + 52x + 1 = 0 => x = -15.63, x = -1.71

These are also called the "critical" points where the derivative is zero.

Coming to other geometrical features of this curve: The y-intercept of this curve is at y=-800.
And the second derivative of this curve becomes zero at x = -8.67. At this point the curve changes concavity.

Example 4

Graph of y = f(x) = (x- 10)(x+20)(x+20) = x³ + 30x² + 0x -4000

A few computed points on the curve, apart from the zero(s) which are known:
(-21,-31), (-18,-112), (-15,-625), (-12,-1408), (-9,-2299), (-6,-3136), (-3,-3757), (0,-4000), (3,-3703), (6,-2704), (9,-841), (12,2048), (15,6125)

Characteristics of the Graph Plot, Curve Sketching of Cubic Curves

If the given cubic function is: f(x) = ax³+ bx²+ cx + d
The derivative of this function is: f'(x) = 3ax² + 2bx + c

The function given to us us f(x) = (x- 10)(x+20)(x+20) = x³ + 30x² + 0x -4000

And the derivative for this is f'(x) = 3x² + 60x + 1

Consider the cubic equation f(x) = (x- 10)(x+20)(x+20) = x³ + 30x² + 0x -4000 = 0

The roots of this cubic equation are at:

(x - (10)) = 0 => x = 10,

OR (x - (-20)) = 0 => x = -20,

OR (x - (-20)) = 0 => x =-20

This cubic equation has a double root at x = -20.
So the plotted cubic curve has not only a zero for that value of x, but the first derivative at that point is also zero. A tangent to the curve drawn at that point will be parallel to the x-axis.

The turning or stationary points is where f'(x) = 0 => 3x² + 60x + 1 = 0 => x = -20.0, x = 0.0

These are also called the "critical" points where the derivative is zero.

Coming to other geometrical features of this curve: The y-intercept of this curve is at y=-4000.
And the second derivative of this curve becomes zero at x = -10.0. At this point the curve changes concavity. A cubic curve has point symmetry around the point of inflection or inflexion.

The zeroes of a polynomial, if they are known, and the coefficients of that polynomial are two different sets of numbers that have interesting relations. If we know the zeroes, then we can write down algebraic expressions for the coefficients. Going the other way is much harder and cannot be done in general.

Example 5

Graph of y = f(x) = (x+20)(x+20)(x+20) = x³ + 60x² + 1200x + 8000

A few computed points on the curve, apart from the zero(s) which are known:
(-21,-1), (-18,8), (-15,125)

Characteristics of the Graph Plot, Curve Sketching of Cubic Curves

If the given cubic function is: f(x) = ax³+ bx²+ cx + d
The derivative of this function is: f'(x) = 3ax² + 2bx + c

The function given to us us f(x) = (x+20)(x+20)(x+20) = x³ + 60x² + 1200x + 8000

And the derivative for this is f'(x) = 3x² + 120x + 1

Consider the cubic equation f(x) = (x+20)(x+20)(x+20) = x³ + 60x² + 1200x + 8000 = 0

The roots of this cubic equation are at:

(x - (-20)) = 0 => x = -20,

OR (x - (-20)) = 0 => x = -20,

OR (x - (-20)) = 0 => x =-20

This cubic equation has a triple root at x = -20.
So the plotted cubic curve has not only a zero for that value of x, but the derivative and second derivative at the point are also zero.

The turning or stationary points is where f'(x) = 0 => 3x² + 120x + 1 = 0 => x = -20.0, x = -20.0

These are also called the "critical" points where the derivative is zero.

Coming to other geometrical features of this curve: What we see here is the graph of a nonlinear function. The y-intercept of this curve is at y=8000.
And the second derivative of this curve becomes zero at x = -20.0. At this point the curve changes concavity. A cubic curve has point symmetry around the point of inflection or inflexion. These are just some of the important features and aspects to keep in mind while trying to visualize and analyze a plot of an algebraic function. A graphical and visual inspection helps in several ways.

The zeroes of a polynomial, if they are known, and the coefficients of that polynomial are two different sets of numbers that have interesting relations. If we know the zeroes, then we can write down algebraic expressions for the coefficients. Going the other way is much harder and cannot be done in general.

Here is a large number of cubic curves which you may study and observe. Curve sketching is an important requirement in many high school exams of Asia, UK, US - and is common, in ISC, IB, CBSE question papers, as well as entrance examinations like the IIT JEE. Curve sketching is a handy tool, used both directly and indrectrly in these examinations.

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