Question

Can I get some help please? Thanks!

Answer 1

`(D)"`

Explanation:

`"the equation of a parabola in "color(blue)"vertex form"` is.

`color(red)(bar(ul(|color(white)(2/2)color(black)(y=a(x-h)^2+k)color(white)(2/2)|)))`
where (h , k ) are the coordinates of the vertex and a is a multiplier.

`• " if "a>0" then minimum turning point " uuu`

`• " if "a<0" then maximum turning point " nnn`

`"here "a=2>0rArr" minimum turning point"`

`y=2(x+2)^2-3" is in vertex form"`

`rArrcolor(magenta)"vertex "=(-2,-3)`

`2(x+2)^2>=0" for all real x"`

`rArr"minimum value "=-3`

`y" is defined for all real values of x"`

`rArr" domain is " x inRR`

`"range is "y inRR,y>=-3`
graph{2(x+2)^2-3 [-10, 10, -5, 5]}

Answer 2

Graph of shape `uu`

`"Minimum "->"Vertex"->(x,y)=(-2,-3)`
Thus `y_("minimum")=-3`

Input `color(white)(.)->" Domain "->x->(-oo,+oo) in RR`
Output`->" Range "color(white)("d")->y->[-3,+oo) in RR-> RR>=-3`

THUS D

Explanation:

Given: `y=2(x+2)^2-3`

This is the vertex form of a quadratic (completing the square).

If you were to expand the bracket your first term would be `+2x^2`

As this is positive the graph is of form `uu` thus the vertex is a minimum.

Consider the standardised form of `y=a(x+color(red)(b/(2a)))^2+color(green)(k)`

Then `color(white)(..)x_("vertex")= (-1)xxcolor(red)(b/(2a))color(white)(..)` and `color(white)(..) y_("vertex")=color(green)(k)`

So we have `"Minimum "->"Vertex"->(x,y)=(-2,-3)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This is one way to remember the relationship

Input `color(white)(..)`comes before Output
D `color(white)("dddd.")`comes before `color(white)(.)` R
Domain comes before Range

Input `color(white)(.)->" Domain "->x->(-oo,+oo) in RR`
Output`->" Range "color(white)("d")->y->[-3,+oo) in RR -> -3<=y`