Let us consider the cubic function f(x) = (x+20)(x+20)(x+14) = x
^{3} + 54x^{2} + 960x + 5600. We will inspect the graph, the zeroes, the turning and inflection points in the cubic curve curve y = f(x).
Cubic Polynomials and EquationsA 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.
Graph of y = f(x) = (x+20)(x+20)(x+14) = x^{3} + 54x^{2} + 960x + 5600A few computed points on the curve, apart from the zero(s) which are known: (-21,-7), (-18,-16), (-15,-25), (-12,128), (-9,605) Characteristics of the Graph Plot, Curve Sketching of Cubic Curves^{3 }+ bx^{2 }+ cx + d The derivative of this function is: f'(x) = 3ax ^{2} + 2bx + cThe function given to us us f(x) = (x+20)(x+20)(x+14) = x
^{3} + 54x^{2} + 960x + 5600And the derivative for this is f'(x) = 3x
^{2} + 108x + 1Consider the cubic equation f(x) = (x+20)(x+20)(x+14) = x
^{3} + 54x^{2} + 960x + 5600 = 0The roots of this cubic equation are at:
(x - (-20)) = 0 => x = -20,
OR (x - (-20)) = 0 => x = -20, OR (x - (-14)) = 0 => x =-14This 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 => 3x
^{2} + 108x + 1 = 0 => x = -20.0, x = -16.0 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=5600.
And the second derivative of this curve becomes zero at x = -18.0. 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. 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
Check the plot of another cubic curve here with roots at -17, -20, -14 Here's another cubic curve here with roots at -20, -10, -14 Here's another cubic curve here with roots at -20, -20, -8 Many of these concepts are a part of the Grade 9,10,11,12 (High School) Mathematics syllabus of the UK GCSE/GCE curriculum, Common Core Standards in the US, ICSE/CBSE/SSC/NTSE syllabus in Indian high schools. You may check out our free and printable worksheets for Common Core and GCSE. |