Answer:
[tex]g(x)=(x-7)^2+6[/tex]
g(x) is a translation of f(x) 7 units to the right and 6 units up.
Step-by-step explanation:
Let the equation of the function g(x) be
[tex]g(x)=a(x-b)^2+c[/tex]
This curve passes through the points (9,10), (5,10) and (7,6), then their coordinates satisfy the equation:
[tex]10=a(9-b)^2+c\\ \\10=a(5-b)^2+c\\ \\6=a(7-b)^2+c[/tex]
Subtract the second equation from the first:
[tex]10-10=a(9-b)^2+c-a(5-b)^2-c\\ \\0=a((9-b)^2-(5-b)^2)\\ \\a\neq 0\ \text{then}\ (9-b)^2-(5-b)^2=0\\ \\(9-b)^2=(5-b)^2\\ \\9-b=5-b\ \text{or}\ 9-b=b-5\\ \\9=5\ \text{false}\\ \\2b=14\\ \\b=7[/tex]
Then
[tex]10=a(9-7)^2+c\\ \\6=a(7-7)^2+c[/tex]
So,
[tex]10=4a+c\\ \\6=c[/tex]
Hence,
[tex]c=6\\ \\b=7\\ \\10=4a+6\Rightarrow a=1[/tex]
The expression for g(x) is
[tex]g(x)=(x-7)^2+6[/tex]
g(x) is a translation of f(x) 7 units to the right and 6 units up.
