Question

How do you find all the critical points to graph `x^2 /9- y^2 /9 =1` including vertices, foci and asymptotes?

Answer 1

The vertices are `( +3,0)` and `(-3,0)`
The foci are `F=(3sqrt2,0)` and `F'=(-3sqrt2,0)`
The asymptotes are `y=x` and `y=-x`

Explanation:

The equation is a left right hyperbola.
`(x-h)^2/a^2-(y-k)^2/b^2=1`
The center is `h,k` `=>` `(0,0)`
The vertices are `(h+-a,k)` `=>`( +3,0)`and`(-3,0)#

The slopes of the asymptotes are `+-b/a` `=+-3/3=+-1`
The equations of the asymptotes are `y=k+-b/a(x-h)`
`:. y=+-x`
To calculate the foci, we need c, `c^2=a^2+b^2`
`:. c=+-sqrt18`
and the foci are `F=(3sqrt2,0)` and `F'=(-3sqrt2,0)`

graph{x^2/9-y^2/9=1 [-12.66, 12.65, -6.33, 6.33]}