Question

The graphs below have the same shape. What is the equation of the red graph?

Answer 1

we have

$g(x)=4-x^{2}$

the vertex of the function g(x) is the point $(0,4)$

the vertex of the function f(x) is the point $(0,0)$

so

the rule of the translation of g(x) to f(x) is

(x,y)------> (x,y-4)

that means

the translation is $4$ units down

therefore

the equation of the function f(x) is

$f(x)=g(x)-4\\f(x)=(4- x^{2})-4 \\f(x)=- x^{2}$

therefore

the answer is

$f(x)=- x^{2}$

Answer 2

g(x)=-x^2 is right for apex