Graph of a Quartic Function with zeros(roots) at [-20, -20, -15, 1]

Let's analyze the following polynomial function.

f(x) = (x+15)(x+20)2
      = x432

You might also be interested in reading about quadraticcubic functions 
Let us analyze the graph of this function which is a quartic polynomial. A quartic polynomial is a fourth degree polynomial. 
  • These have zero to four real roots
  • They may have one, two or three extrema (maxima/minima - turning points)
  • Zero, one or two inflection (inflexion) points.
Let's do some curve sketchingy = f(x) = (x+15)(x+20)2= x432

We can plot the curve by computing some sample points other than the zeroes already known. 
Some sample points are: A few computed points on the curve, apart from the zero(s) which are known:

Quartic Polynomial Curve Graph

Let us inspect the roots of the given polynomial function.

These values of x are the roots of the quadratic equation (x+15)(x+20)2

Let us analyze the turning points in this curve. 
The derivative of the given function = f'(x) = 4x32
This is a cubic function. 
Turning points are those where the derivative of the function (obtained on differentiating the function) is zero. 
These may be obtained by solving the cubic equation 4x32
The turning points of this curve are approximately at x = [-20.0, -16.7, -3.7]. At these points, the curve has either a local maxima or minima. These are the extrema - the peaks and troughs in the graph plot..

Now, let us take a look at the inflexion
These are the points where the curve changes concavity. 
The second derivative f'''(x) = 12x32
This is a quadratic function. The value(s) of x for which this quadratic function is zero, are the inflexion points. 
This curve has two inflexion points at (approx) -18.47 and -8.53. At these points, the concavity changes.

Check the plot of another quartic curve here
Here's another quartic curve here
Here's another quartic curve here
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Prashant Bhattacharji,
Mar 20, 2017, 12:26 AM