Question

How do you find the vertex and intercepts for `y = x^2 - 8x + 18`?

Answer 1

Vertex={(4,2)}, intercept={(0,18)}

Explanation:

`y=x^2-8x+18`
`=>y'=2x-8`
For `y'=0`,
`2x-8=0`
`=>x=4`
`=>y=4^2-8*4+18`
`=16-32+18`
`=2`
`=> vertex={(4,2)}`
`Delta=(-8)^2-4*1*18`
`=-8`
`=> Delta<0`
For `y=0`,
`x^2-8x+18=0`
`=>` `x` is imaginary
For `x=0`,
`y=18`
`=>` intercept={(0,18)}

Answer 2

vertex: `(4,2)`
y-intercept: `18`
there is no x-intercept

Explanation:

Converting the given equation into vertex form: `y=m(x-a)^2+b` with vertex at `(a,b)`

`y=x^2-8x+18`

`rarr y=x^2-8x+16+2`

`rarr y=1(x-4)^2+2`
`color(white)("XXX")with vertex at`(4,2)#

y-intercept is the value of `y` when `x=0`
`rArr` y-intercept = 18#

x-intercept is the value of `x` when `y=0`
i.e. when `x^2-8x+18=0`
but checking the discriminant (`b^2-4ac` using the standard form)
we see that there are no solutions for `x` since the discriminant is `< 0`