Question 1000272


As with any function, the domain of a quadratic function f(x) is the set of x-values for which the function is defined, and the range is the set of all the output values (values of f).

Quadratic functions generally have the whole real line as their domain: 
any {{{x}}} is a legitimate input. 
The range is {{{restricted}}} to those points greater than or equal to the {{{y-coordinate}}} of the {{{vertex}}} (or less than or equal to, depending on whether the parabola opens {{{up}}} or {{{down}}}). 

you are given {{{x^2-2x-3=0}}} ...since coefficient {{{a=1}}} which is positive number, your parabola opens {{{up}}}

The equation for a parabola can also be written in "vertex form":

{{{y = a(x - h)^2 + k}}}
{{{y=x^2-2x-3}}}

{{{y=(x^2-2x+b^2)-b^2-3}}}

{{{y=(x^2-2x+1^2)-1^2-3}}}
{{{y=(x-1)^2-1-3}}}
{{{y=(x-1)^2-4}}}

=>{{{h=1}}} and {{{k=-4}}}
the {{{vertex}}} of your parabola is at ({{{1}}},{{{-4}}})

so, the range will be all values of {{{y}}} from {{{-4}}} to {{{infinity}}}

or { {{{y}}} element of {{{R}}}: {{{y>=-4}}} }

in interval notation:

[{{{-4}}},{{{infinity}}})


 {{{ graph( 600, 600, -10, 10, -10, 10, (x-1)^2-4) }}}