Second order polynomials are parabolas

The relation definitions of these types of polynomials is:

\[ \{(x,y)\in\mathbb{R}^2\,\,|\,\, y = a_2 x^2+a_1 x+a_0\} \tag{1}\]

for some real parameters \(a_0\), \(a_1\) and \(a_2\), having \(a_2 \not = 0\) guarantees this is a second order polynomial.

Here are some choices of parameters and the corresponding graphs:

The graph above exhibits the windows on the set Equation 1 where interesting behavior of these functions occur, by interesting, I mean, we see see the x-axis and y-axis intersections as well as minima and maxima; outside of the picture, these function either increase or decrease monotonically toward infinity, that’s uninteresting since nothing else happens there.

To know these function even better it is useful to compute by hand (analytically) some important characteristics:

• y-axis intercept: we want to know the coordinates on Equation 1 of the form \((0,y_0)\).

  \[ (0,y_0)\in \{(x,y)\in\mathbb{R}^2\,\,|\,\, y = a_2 x^2+a_1 x+a_0\} \iff y = a_2 \times 0 +a_1\times 0+ a_0 \iff y=a_0 \]

  Thus the point \((0,a_0)\) is the interception of the polynomial with the vertical axis.

• x-axis intercept: seeking point of the form \((x_0,0)\) requires us to solve the equation

  \[ (x_0,0)\in \{(x,y)\in\mathbb{R}^2\,\,|\,\, y = a_2 x^2+a_1 x+a_0\}\iff 0 = a_2 x_0^2+a_1 x_0+a_0 \]

  To solve the second order equation we may use the Resolvent Formula if a solution exists. More on this later.

• when \(x\) is large (far from \(1\)): the process of progressively increasing \(x\) and seeing what happens to the corresponding \(y\)’s.

  The idea is to “solve” the dynamical problem:

  \[ \begin{cases}y = a_2 x^2+a_1 x+a_0\\ x\longrightarrow \pm\infty \end{cases} \tag{2}\]

  by computing how \(y\) behaves when \(x\) increases without bound.

  We solve this problem using our knowledge of how the atomic polynomials behave individually, see the last ordering relation in ?@eq-relations_of_atoms. We know that when \(x\) is far from \(1\) we must have:

  \[ a_0 \ll a_1 x \ll a_2x^2 \]

  Since \(a_2x^2\) term is much larger than \(a_1 x\) and \(a_0\), we can drop them from the equation \(y=a_2x^2+a_1 x+a_0\) , doing so giving us an approximate equation, \(y \sim a_2 x^2\), which is simple and thus easy to graph. The graph of both functions is very similar.

  Exact equation for all \(x\)	Approximate equation when \(x\) is large
  \(y = a_2 x^2+a_1 x+a_0\)	\(y \sim a_2 x^2\)

  That means, if you know how to solve

  \[ \begin{cases}y \sim a_2 x^2\\x\longrightarrow \pm\infty \end{cases} \tag{3}\]

  you know how to solve Equation 2 . But Equation 3 is easy, since it is just an atomic polynomial: if \(x\) gets larger and larger then \(y\) gets larger and larger, the solution is \(y\longrightarrow +\infty\), and we conclude the same thing must happen in Equation 2. Problem solved, but solving a similar but simpler problem!