Question

The graph shows g(x), which is a translation of f(x) = x^2. Write the function rule for g(x). Show step by step.

Points on nonlinear graph are (9,10) (5,10) (7,6).

Answer 1

`g(x)=(x-7)^2+6`

g(x) is a translation of f(x) 7 units to the right and 6 units up.

Explanation:

Let the equation of the function g(x) be

`g(x)=a(x-b)^2+c`

This curve passes through the points (9,10), (5,10) and (7,6), then their coordinates satisfy the equation:

`10=a(9-b)^2+c\\ \\10=a(5-b)^2+c\\ \\6=a(7-b)^2+c`

Subtract the second equation from the first:

`10-10=a(9-b)^2+c-a(5-b)^2-c\\ \\0=a((9-b)^2-(5-b)^2)\\ \\a\\eq 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`

Then

`10=a(9-7)^2+c\\  \\6=a(7-7)^2+c`

So,

`10=4a+c\\  \\6=c`

Hence,

`c=6\\ \\b=7\\ \\10=4a+6\Rightarrow a=1`

The expression for g(x) is

`g(x)=(x-7)^2+6`

g(x) is a translation of f(x) 7 units to the right and 6 units up.