Definition of the Cubic Function

 

Definition of the Cubic Function

The cubic function is a function or graph (curve) is defined by the two formulas below:

a) y= ax^3+bx^2+cx+d    (expanded standard form)

b) y= (x+1)(x+2)(x+3)     (factorized form)  (factors of the cubic function) 

The cubic curve or function has three x intercepts. For the  factorized cubic function as stated earlier y= (x+1)(x+2)(x+3) the x intercepts are x=-1, x=-2 and x= -3. The relationship between the two cubic formulas as stated above is based on the expansion of the factorized cubic formula, y= (x+1)(x+2)(x+3).

The expansion of the cubic function y= (x+1)(x+2)(x+3) is calculated thus:      

y= (x+1)[(x(x+3) +2(x+1)]

y= (x+1)[x^2+3x+2x+1]

y= (x+1)[x^2+5x+1]

y= x[x^2+5x+1]+1[x^2+5x+1]

 y= [x^3+5x^2+x]+x^2+5x+1]

add like terms

 y= [x^3+5x^2+x^2+x+5x+1]

y=......................................................

The intercepts of the cubic functions are evaluated from the factorized form by equating each factor to zero. For the cubic function, y= (x+1)(x+2)(x+3) the x intercepts of the function are as follows:

x+1= 0  

x=-1

x+2=0

x=-2 

x+3=0

x=-3

The x intercepts as evaluated above may be applied in the factor theorem in order to get a zero value as follows:

Substitute the values of x above into the formula 

 y= [x^3+6x^2+6x+1]

f(-3)=....................................................

f(-2)=....................................................

f(-1)=....................................................

Classification of  Cubic Functions or Curves

The nature or value of the x intercepts or roots

The roots of a cubic function may be real values or non real values. A real value or real root is a value such as x= 0, x=+1, x=-1, x=+2 etc. These are values that can be seen on the cartesian plane.

The cubic function y= (2x-7)(3x-1)x, is a function having one real x intercept and two unreal values x=+(1/3) and x=+(7/2). Another cubic function in this category is y= (3x+1)(2x+9)(2x+3). 

The values of at least one or two of the coefficients of the formula y= ax^3+bx^2+cx+d is equal to zero (d=0, c=0, b=0 and a is not equal to zero etc)

Examples of these cubic functions are y=x^3, y=x^2(x-7), y=x(x^2-7), y=x^2(3x-7) and y=x(x^2-10).

For the cubic functions  y=x^3, a=1, b=0, c=0 and d=0. This is not the case for the other cubic functions as stated above. One of the roots of the functions y=x^2(x-7), y=x(x^2-7), y=x^2(3x-7) and y=x(x^2-10) is x=0. This correct based on the fact that d=0. The values of a, b and c are not equal to zero.

Maximum Point and Minimum Point

The x values of the maximum point, minimum point and the point of inflexion are three different x coordinates. This based on the first derivative of the cubic function equals to zero. The application of the quadratic formula  will give us two values of x.

Another perspective of this criterion is the maximum point, minimum point and point of inflexion intersecting at one and the same point. This is correct for cubic functions such as y=x^3, y=(x+3)^3 and y=(2x-5)^3. This also means that the maximum point, minimum point and the point of inflexion are not three different x coordinates.  

Repeated root

The  cubic functions y= x(x+2)^2 and y= x^2(x+2) have the repeated roots or x intercepts x=-2 and x=0 respectively. The relationship between the value of the x intercepts and the factor of the cubic function lies in the factor that has the exponent two (or squared). Usually, this x intercept is a turning point and an x intercept.

The Positive or Negative value of a in the cubic function y= ax^3+bx^2+cx+d.

The Figure 3 illustrates the relationship between the positive or negative value of a and the plotting of a cubic curve in terms of maximum point and minimum point as we progress from the negative x axis to the positive x axis.

Refer to Fig 3 and Fig 1. The minimum point is plotted before the maximum point as we progress from the negative x axis to the positive x axis. The reverse is the case for the function y= x^2(x+2).

The classification of the cubic function is based on at least any two or three criteria above.

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