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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 :
$$f\left(x\right)=a{x}^{2}+bx+c;\phantom{\rule{1em}{0ex}}a,b,c\in R,\phantom{\rule{1em}{0ex}}a>0$$
Quadratic equation is obtained by equating quadratic function to zero. General form of quadratic equation corresponding to quadratic function is :
$$a{x}^{2}+bx+c=0;\phantom{\rule{1em}{0ex}}a,b,c\in R,\phantom{\rule{1em}{0ex}}a>0$$
Nature of a given quadratic function is best understood in terms of discriminant, D, of corresponding quadratic equation. This is given as :
$$D={b}^{2}-4ac$$
Quadratic equation is obtained by equating quadratic function to zero. Quadratic equation has at most two roots. The roots are given by :
$$\alpha =\frac{-b-\sqrt{D}}{2a}=\frac{-b-\sqrt{{b}^{2}-4ac}}{2a}$$
$$\beta =\frac{-b+\sqrt{D}}{2a}=\frac{-b+\sqrt{{b}^{2}-4ac}}{2a}$$
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.
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 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 ${b}^{2}/4{a}^{2}$ as :
$$\Rightarrow y=a\left({x}^{2}+\frac{b}{a}x+\frac{{b}^{2}}{4{a}^{2}}+\frac{c}{a}-\frac{{b}^{2}}{4{a}^{2}}\right)$$
$$\Rightarrow y=a\{{\left(x+\frac{b}{2a}\right)}^{2}-\frac{{b}^{2}-4ac}{4a}\}$$
$$\Rightarrow y+\frac{{b}^{2}-4ac}{4a}=a{\left(x+\frac{b}{2a}\right)}^{2}$$
$$\Rightarrow y+\frac{D}{4a}=a{\left(x+\frac{b}{2a}\right)}^{2}$$
$$\Rightarrow Y=a{X}^{2}$$
Where,
$$X=x+\frac{b}{2a}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}Y=y+\frac{D}{4a}$$
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.
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 :
$${y}_{\mathrm{min}}=-\frac{D}{4a}\phantom{\rule{1em}{0ex}}\text{at}\phantom{\rule{1em}{0ex}}x=-\frac{b}{2a}$$
$${y}_{\mathrm{max}}\Rightarrow \infty $$
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 :
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