ANSWER
Option C.
EXPLANATION
The first equation is
[tex]y = x - 2[/tex]
We can easily graph this straight line because it has a slope of 1 and a y-intercept of -2.
The second equation is
[tex]y = {x}^{2} - 6x + 8[/tex]
This is a graph of a quadratic function. If we write this in vertex form, we can easily graph it using transformations.
We complete the square to get the function to the vertex form as follows:
[tex]y = {x}^{2} - 6x +9 + 8 - 9[/tex]
[tex]y = (x { - 3)}^{2} - 1[/tex]
This quadratic graph has its minimum point (vertex) at (3,-1).
Points of intersection
To find the point of intersection of the two graphs, we equate the two functions and solve for x.
[tex]{x}^{2} - 6x + 8 = x - 2[/tex]
We rewrite in standard form
[tex] {x}^{2} - 7x + 10 = 0[/tex]
Factor to obtain
[tex](x - 2)(x - 5) = 0[/tex]
The solutions are
[tex]x = 2 \: and \: x = 5[/tex]
When x=2, y=2-2=0
This gives (2,0) as a point of intersection.
When x=5, y=5-2=3.
This gives us (5,3) as another point of intersection.
The correct answer is option C.
