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$${y}_{\mathrm{max}}=-\frac{D}{4a}\phantom{\rule{1em}{0ex}}\text{at}\phantom{\rule{1em}{0ex}}x=-\frac{b}{2a}$$
$${y}_{\mathrm{min}}\Rightarrow -\infty $$
Clearly, range of the function is (-∞, -D/4a].
Problem : Determine range of $f\left(x\right)=-3{x}^{2}+2x-4$
Solution : The determinant of corresponding quadratic equation is :
$$D={b}^{2}-4ac=4-4X\left(-3\right)X\left(-4\right)=4-48=-44\phantom{\rule{1em}{0ex}}\Rightarrow D<0$$
$$a=-3\phantom{\rule{1em}{0ex}}\Rightarrow a<0$$
The graph of function is parabola opening down. Its vertex represents the maximum function value. The maximum and minimum values of function are given by :
$$\Rightarrow {y}_{\mathrm{max}}=-\frac{D}{4a}=-\frac{-44}{4X-3}=-\frac{44}{12}=-\frac{11}{3}$$
$${y}_{\mathrm{min}}\Rightarrow -\infty $$
Range = (-∞, -11/3)
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.
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.
$$\phantom{\rule{1em}{0ex}}D<\mathrm{0,}a>0\phantom{\rule{1em}{0ex}}\Rightarrow f\left(x\right)>0\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}x\in R$$
If a<0, then parabola opens down. The value of quadratic function is negative for all values of x i.e.
$$\phantom{\rule{1em}{0ex}}D<\mathrm{0,}a<0\phantom{\rule{1em}{0ex}}\Rightarrow f\left(x\right)<0\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}x\in R$$
Sign rule : If D<0, then sign of function is same as that of “a” for all values of x in R.
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.
$$D=\mathrm{0,}a>0\phantom{\rule{1em}{0ex}}\Rightarrow f\left(x\right)\ge 0\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}x\in R$$
If a<0, then parabola opens down. The value of quadratic function is non-positive for all values of x i.e.
$$D=\mathrm{0,}a<0\phantom{\rule{1em}{0ex}}\Rightarrow f\left(x\right)\le 0\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}x\in R$$
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.
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).
$$D>\mathrm{0,}a>0\phantom{\rule{1em}{0ex}}\Rightarrow f\left(x\right)>0\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}x\in \left(-\infty ,\alpha \right)\cup \left(\beta ,\infty \right)\phantom{\rule{1em}{0ex}}\Rightarrow \text{Sign of function same as that of \u201ca\u201d}$$
$$D>\mathrm{0,}a>0\phantom{\rule{1em}{0ex}}\Rightarrow f\left(x\right)=0\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}x\in \{\alpha ,\beta \}\phantom{\rule{1em}{0ex}}$$
$$D>\mathrm{0,}a>0\phantom{\rule{1em}{0ex}}\Rightarrow f\left(x\right)<0\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}x\in \left(\alpha ,\beta \right)\phantom{\rule{1em}{0ex}}\Rightarrow \text{Sign of function opposite to that of \u201ca\u201d}$$
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,∞).
$$D>\mathrm{0,}a<0\phantom{\rule{1em}{0ex}}\Rightarrow f\left(x\right)<0\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}x\in \left(\alpha ,\beta \right)\phantom{\rule{1em}{0ex}}\Rightarrow \text{Sign of function same as that of \u201ca\u201d}$$
$$D>\mathrm{0,}a<0\phantom{\rule{1em}{0ex}}\Rightarrow f\left(x\right)=0\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}x\in \{\alpha ,\beta \}\phantom{\rule{1em}{0ex}}$$
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