Graph of Cubic Functions/Cubic Equations for zeros and roots (-2,-10,-8)

Let us consider the cubic function f(x) = (x+2)(x+10)(x+8) = x3 + 20x2 + 116x + 160. 

Cubic Polynomials and Equations

A cubic polynomial is a polynomial of degree 3.
where a is nonzero.

Graph of y = f(x) = (x+2)(x+10)(x+8) = x3 + 20x2 + 116x + 160
Cubic Polynomial Curve Plot on Graph

(-11,-27), (-8,0), (-5,-45), (-2,0), (1,297)

Characteristics of the Graph Plot, Curve Sketching of Cubic Curves

And the derivative for this is f'(x) = 3x2 + 40x + 1

Consider the cubic equation f(x) = (x+2)(x+10)(x+8) = x3 + 20x2 + 116x + 160 = 0
The roots of this cubic equation are at: 
(x - (-2)) = 0 => x = -2, 
This cubic equation has real and unique roots at -2, -10, -8.

The turning or stationary points is where f'(x) = 0 => 3x2 + 40x + 1 = 0 => x = -9.07, x = -4.26  
And the second derivative of this curve becomes zero at x = -6.67. 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.


Compute the Area between the given curve and the X-Axis

F(x) = Integral of f(x) = ∫ ( x3 + 20x2 + 116x + 160) dx = x4/3 + 20x3/2 + 116x2 + 160x + C
Where C is the constant of integration.
Values of F at each of the roots -10,-8,-2:
F(-10) = (-10)4/3 + (20 * (-10)3)/2 + 116*(-10)2 + 160*(-10) + C = 3333.33 + C
Similarly, we can compute: F(-8) = 2389.33 + C and F(-2) = 69.33 + C
Where C is the constant of integration.

Area between the curve and the X-axis = |Area enclosed for x in interval (-10,-8)| + |Area enclosed for x in interval (-8,-2)|
= |Definite integral of f(x) between limits -10 and -8| + |Definite integral of f(x) between limits -8 and -2|
= |F(-8) - F(-10)| + |F(-2) - F(-8)|
= |2389.33 + C - (3333.33 + C)| + |69.33 + C - (2389.33 + C)|
Terms involving the constant of integration cancel out.
Area = |-944.0| + |-2320.0|=> Area = 1376.0 Square Units

Check the plot of another cubic curve here with roots at 1, -10, -8

Here's another cubic curve here with roots at -2, 0, -8

Here's another cubic curve here with roots at -2, -10, -2

Prashant Bhattacharji,
Mar 21, 2017, 1:36 AM