Answer: The vertex is (3, 16) and the x-intercepts are (-1, 0) and (7, 0).
Step-by-step explanation: We are given to find the vertex and x-intercepts of the graph of the following function :
[tex]y=x^2-6x-7~~~~~~~~~~~~~~~~~~~~~~~~~~~`(i)[/tex]
We know that
the vertex of the graph of function [tex]y=a f(x-h)^2-k[/tex] is given by (h, k).
From equation (i), we have
[tex]y=x^2-6x-7\\\\\Rightarrow y=(x^2-6x+9)-7-9\\\\\Rightarrow y=(x-3)^2-16.[/tex]
Therefore, the vertex is (3, 16).
The x-intercepts of function (i) will be given by
[tex]x^2-6x-7=0\\\\\Rightarrow x^2-7x+x-7=0\\\\\Rightarrow x(x-7)+1(x-7)=0\\\\\Rightarrow (x+1)(x-7)=0\\\\\Rightarrow x+1=0,~~x-7=0\\\\\Rightarrow x=-1,~~x=7.[/tex]
Thus, the vertex is (3, 16) and the x-intercepts are (-1, 0) and (7, 0).
