Question 117705
{{{y(x) = x^2 - 6x + 7}}}
Find roots
{{{x^2 - 6x + 7 = 0}}}
{{{x^2 - 6x = -7}}}
complete the square by taking half of the {{{-6}}}, square it, 
then add it to both sides
{{{x^2 - 6x + 9 = -7 + 9}}}
{{{(x - 3)^2 = 2}}}
{{{x - 3 = 0 +-sqrt(2)}}}
{{{x = 3 + sqrt(2)}}}
{{{x = 3 - sqrt(2)}}}
The roots (x-intercepts) are at (3 + {{{sqrt(2)}}}, 0) and (3 - {{{sqrt(2)}}},0)
The vertex is exactly between them at (3, y). What is {{{y}}}?
{{{y(x) = x^2 - 6x + 7}}} make {{{x=3}}}
{{{y(3) = 3^2 - 6*3 + 7}}}
{{{y(3) = 9 - 18 + 7}}}
{{{y(3) = -2}}}
So, the vertex is at (3,-2)
Lastly, where is the y-intercept? It is where {{{x=0}}}
{{{y(x) = x^2 - 6x + 7}}}
{{{y(0) = 0^2 - 6*0 + 7}}}
{{{y(0) = 7}}}
The y-intercept is at (0,7)
Here's the graph
{{{ graph( 600, 600, -10, 10, -10, 10, x^2 - 6x + 7) }}}