Question

How do you write `y=x^2+8x+14` in vertex form and identify the vertex, y intercept and x intercept?

Answer 1

`"see explanation"`

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"`

`"to obtain this form "color(blue)"complete the square"`

`y=x^2+2(4)xcolor(red)(+16)color(red)(-16)+14`

`rArry=(x+4)^2-2larrcolor(blue)"in vertex form"`

`rArrcolor(magenta)"vertex "=(-4,-2)`

`"to obtain the intercepts"`

`• " let x = 0, in the equation for y-intercept"`

`• " let y = 0, in the equation for x-intercept"`

`x=0rArry=0+0+14=14larrcolor(red)"y-intercept"`

`y=0rArr(x+4)^2-2=0larr"add 2 to both sides"`

`rArr(x+4)^2=2`

`color(blue)"take the square root of both sides"`

`sqrt((x+4)^2)=+-sqrt2larrcolor(blue)"note plus or minus"`

`rArrx+4=+-sqrt2larr"subtract 4 from both sides"`

`rArrx=-4+-sqrt2larrcolor(red)"exact values"`
graph{(y-x^2-8x-14)((x+4)^2+(y+2)^2-0.04)=0 [-10, 10, -5, 5]}