3.3 Quadratic polynomial function

We are already acquainted with quadratic equation and its roots. In this module, we shall study quadratic expression from the point of view of a function. It is a polynomial function of degree 2. The general form of quadratic expression/ function is :

$$f\left(x\right)=a{x}^2+bx+c;a,b,c\in R,a>0$$

Elements of quadratic equation

Quadratic equation

Quadratic equation is obtained by equating quadratic function to zero. General form of quadratic equation corresponding to quadratic function is :

$$a{x}^2+bx+c=0;a,b,c\in R,a>0$$

Discriminant of quadratic equation

Nature of a given quadratic function is best understood in terms of discriminant, D, of corresponding quadratic equation. This is given as :

$$D=b^2-4ac$$

Roots of quadratic equation

Quadratic equation is obtained by equating quadratic function to zero. Quadratic equation has at most two roots. The roots are given by :

$$\alpha =\frac{-b-\sqrt{D}}{2a}=\frac{-b-\sqrt{b^2-4ac}}{2a}$$

$$\beta =\frac{-b+\sqrt{D}}{2a}=\frac{-b+\sqrt{b^2-4ac}}{2a}$$

Properties of roots of quadratic equation

1 : If D>0, then roots are real and distinct.

2 : If D=0, then roots are real and equal.

3 : If D<0, then roots are complex conjugates with non-zero imaginary part.

4 : If D>0; a,b,c∈T (rational numbers) and D is a perfect square, then roots are rational.

5 : If D>0; a,b,c∈T (rational numbers) and D is not a perfect square, then roots are radical conjugates.

6 : If D>0; a=1;b,c∈Z (integer numbers) and roots are rational, then roots are integers.

7 : If a quadratic equation has more than two roots, then the function is an identity in x and a=b=c=0.

8 : If a quadratic equation has one real root and a,b,c∈R, then other root is also real.

Elements of quadratic function

Zeroes of quadratic function

The real roots of the quadratic equation are zeroes of quadratic function. The zeroes of quadratic function are real values of x for which value of quadratic function becomes zero. On graph, zeros are the points at which graph intersects y=0 i.e. x-axis.

Graph of quadratic function

Graph reveals important characteristics of quadratic function. The graph of quadratic function is a parabola. Working with the quadratic function, we have :

$$y=a{x}^2+bx+c=a\left(x^2+\frac{b}{a}x+\frac{c}{a}\right)$$

In order to complete square, we add and subtract $b^2/4a^2$ as :

$$\Rightarrow y=a\left(x^2+\frac{b}{a}x+\frac{b^2}{4a^2}+\frac{c}{a}-\frac{b^2}{4a^2}\right)$$

$$\Rightarrow y=a\{{\left(x+\frac{b}{2a}\right)}^2-\frac{b^2-4ac}{4a}\}$$

$$\Rightarrow y+\frac{b^2-4ac}{4a}=a{\left(x+\frac{b}{2a}\right)}^2$$

$$\Rightarrow y+\frac{D}{4a}=a{\left(x+\frac{b}{2a}\right)}^2$$

$$\Rightarrow Y=a{X}^2$$

Where,

$$X=x+\frac{b}{2a}\text{and}Y=y+\frac{D}{4a}$$

Graph of quadratic function

Clearly, $Y=a{X}^2$ is an equation of parabola having its vertex given by (-b/2a, -D/4a). When a>0, parabola opens up and when a<0, parabola opens down. Further, parabola is symmetric about x=-b/2a.

Maximum and minimum values of quadratic function

The graph of quadratic function extends on either sides of x-axis. Its domain, therefore, is R. On the other hand, value of function extends from vertex to either positive or negative infinity, depending on whether “a” is positive or negative.

When a>0, the graph of quadratic function is parabola opening up. The minimum and maximum values of the function are given by :

$$y_{\min }=-\frac{D}{4a}\text{at}x=-\frac{b}{2a}$$

$$y_{\max }\Rightarrow \infty$$

Clearly, range of the function is [-D/4a, ∞).

When a<0, the graph of quadratic function is parabola opening down. The maximum and minimum values of the function are given by :