Polynomial functions: constant, affine and quadratic

A polynomial function is a function whose analytic expression is given by a polynomial: $$f(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0$$ with $n \in \mathbb{N}\cup \{0\}$, $a_n,a_{n-1},\ldots, a_1,,_a0 \in \mathbb{R}$ and $a_n\neq 0$ if $n\neq 0$.

Since the polynomials can be evaluated in any real number, we have that the domain of the polynomial functions is $\mathbb{R}$, that is $Dom(f)=\mathbb{R}$.

The image of this type of functions is not always clear:

• Polynomials of odd degree: This is the simplest case since $Im(f)=\mathbb{R}$.
• Polynomial of even degree: The image will depend on the coefficients of the polynomial, which will determine its orientation and its relative extrema. In the case $n = 2$, which we will call quadratic functions, it is enough to know the vertex of the parabola and to take into account the sign of the first coefficient.

Constant function: $f (x) = k$

This is a polynomial of degree $0$. Its graph is a horizontal straight line that goes along the point where $y=k$ (and therefore $Im (f) = k$).

An example of constant function is $f (x) =-1$:

Affine function: $f (x) = ax + b$

In order to have a linear function we need $a\neq 0$. The degree of this function is $1$. Its graph is a straight line that goes through the point $(0, b)$ and its slope depends on the value of $a$.

In the particular case in which $b = 0$, we will call it a linear function: $f (x) = ax$. This function is equivalent to the function of direct proportionality, where $a$ is the proportionality constant.

In the particular case in which $a = 1$, we obtain the identity function, that is, $f (x) = x$ , whose graph is the $45$ degree line.

An example of affine function is $f (x) = 3x - 1$.

Quadratic function:$f (x) = ax^2 + bx + c$

To be a quadratic function it is necessary to have $a\neq 0$. This is a function of second degree, whose graph is an $\cup$ shaped parabola if $a> 0$, or an $\cap$ shaped parabola if $a <0$ (see the unit on convexity and concavity).

The vertex of the above mentioned parabola is $\displaystyle \Big(-\frac{b}{2a}, -\frac{b^2-4ac}{4a}\Big)$.

The intersection point with the vertical axis is $c$. The points cutting through the horizontal axis are the solutions to the equation of the second degree if they exist.

An example of quadratic function is $f(x) =x^2-2x+1$.