3.3 Quadratic polynomial function  (Page 2/3)

Graph of quadratic function

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

$$y_{\min }\Rightarrow -\infty$$

Clearly, range of the function is (-∞, -D/4a].

Nature of quadratic function

The discriminant of corresponding quadratic equation and coefficient of term “ $x^2$ ” of quadratic function together determine nature of quadratic function and hence its graph. Graphs of quadratic function is intuitive and helpful to remember results. As a matter of fact, we can interpret all properties of quadratic function, if we can draw its graph.

Case 1 : d<0

If D<0, then roots are complex conjugates. It means graph of function does not intersect x-axis. If a>0, then parabola opens up. The value of quadratic function is positive for all values of x i.e.

$$D<\mathrm{0,}a>0\Rightarrow f\left(x\right)>0\text{for}x\in R$$

Graph of quadratic function

If a<0, then parabola opens down. The value of quadratic function is negative for all values of x i.e.

$$D<\mathrm{0,}a<0\Rightarrow f\left(x\right)<0\text{for}x\in R$$

Sign rule : If D<0, then sign of function is same as that of “a” for all values of x in R.

Case 2 : d=0

If D=0, then roots are equal and is given by –b/2a. It means graph of function just touches x-axis. If a>0, then parabola opens up. The value of quadratic function is non-negative for all values of x i.e.

$$D=\mathrm{0,}a>0\Rightarrow f\left(x\right)\ge 0\text{for}x\in R$$

Graph of quadratic function

If a<0, then parabola opens down. The value of quadratic function is non-positive for all values of x i.e.

$$D=\mathrm{0,}a<0\Rightarrow f\left(x\right)\le 0\text{for}x\in R$$

Sign rule : If D=0, then sign of function is same as that of “a” for all values of x in R except at x=-b/2a, at which f(x)=0. We do not associate sign with zero.

Case 3 : d>0

If D>0, then roots are unequal and are given by (–b±D)/2a. It means graph of function intersects x-axis at α and β (β>α). If a>0, then parabola opens up. The value of quadratic function is positive for all values of x in the interval (-∞,α) U (β,∞).The values of quadratic function are zero for values of x ∈{α,β}. The value of quadratic function is negative for all values of x in the interval (α,β).

Graph of quadratic function

$$D>\mathrm{0,}a>0\Rightarrow f\left(x\right)>0\text{for}x\in \left(-\infty ,\alpha \right)\cup \left(\beta ,\infty \right)\Rightarrow \text{Sign of function same as that of “a”}$$

$$D>\mathrm{0,}a>0\Rightarrow f\left(x\right)=0\text{for}x\in \{\alpha ,\beta \}$$

$$D>\mathrm{0,}a>0\Rightarrow f\left(x\right)<0\text{for}x\in \left(\alpha ,\beta \right)\Rightarrow \text{Sign of function opposite to that of “a”}$$

If a<0, then parabola opens down. The value of quadratic function is positive for all values of x in the interval (α,β).The values of quadratic function are zero for values of x ∈{α,β}. The value of quadratic function is negative for all values of x in the interval (-∞,α) U (β,∞).

Graph of quadratic function

$$D>\mathrm{0,}a<0\Rightarrow f\left(x\right)<0\text{for}x\in \left(\alpha ,\beta \right)\Rightarrow \text{Sign of function same as that of “a”}$$

$$D>\mathrm{0,}a<0\Rightarrow f\left(x\right)=0\text{for}x\in \{\alpha ,\beta \}$$