Question

How do you tell whether the graph opens up or down, find the vertex, and find the axis of symmetry of `y=(x+3)(x+1)`?

Answer 1

`"see explanation"`

Explanation:

`"for a quadratic in standard form " y=ax^2+bx+c`

`• " if " a>0" then graph opens up " uuu`

`• " if " a<0" then graph opens down " nnn`

`y=(x+3)(x+1)`

`"expand the factors using FOIL"`

`rArry=x^2+4x+3larrcolor(blue)" in standard form"`

`"with " a=1,b=4,c=3`

`"since " a>0" then graph opens up"`

`"the x-coordinate of the vertex " x_(color(red)"vertex")=-b/(2a)`

`rArrx_(color(red)"vertex")=-4/2=-2`

`"substitute into equation for y-coordinate of vertex"`

`rArry_(color(red)"vertex")=(-2)^2+4(-2)+3=-1`

`rArrcolor(magenta)"vertex "=(-2,-1)`

`"the axis of symmetry passes through the vertex is a"`
`"vertical line with equation"`

`x=-2`
graph{(y-x^2-4x-3)(y-1000x-2000)=0 [-10, 10, -5, 5]}