the graph shows ( g(x) ), which is a translation of ( f(x)=x^{2} ). write the function rule for ( g(x)…

the graph shows ( g(x) ), which is a translation of ( f(x)=x^{2} ). write the function rule for ( g(x) ).\nwrite your answer in the form ( a(x - h)^{2}+k ), where ( a ), ( h ), and ( k ) are integers or simplified\nfractions.\n( g(x)= )

the graph shows ( g(x) ), which is a translation of ( f(x)=x^{2} ). write the function rule for ( g(x) ).\nwrite your answer in the form ( a(x - h)^{2}+k ), where ( a ), ( h ), and ( k ) are integers or simplified\nfractions.\n( g(x)= )

Answer

Explanation:

Step1: Identify the vertex of the parabola

The vertex form of a parabola is (y = a(x - h)^2 + k), where ((h,k)) is the vertex. From the graph, the vertex is ((-6, - 4)), so (h=-6) and (k = - 4).

Step2: Use a point on the parabola to find (a)

The original function (f(x)=x^2) is transformed. Let's use the point ((0,-1)) on (g(x)). Substitute (x = 0), (y=-1), (h=-6), and (k=-4) into (y=a(x - h)^2 + k): (-1=a(0-(-6))^2+(-4)) (-1=a(6)^2-4) (-1 = 36a-4) Add (4) to both sides: (36a=3), then (a=\frac{3}{36}=\frac{1}{12})

Answer:

(g(x)=\frac{1}{12}(x + 6)^2-4)