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) ).\n\nwrite your answer in the form ( a(x - h)^{2}+k ), where ( a ), ( h ), and ( k ) are integers or simplified fractions.

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

Answer

Explanation:

Step1: Identify the vertex form of a parabola

The vertex form of a parabola is (y = a(x - h)^2 + k), where ((h,k)) is the vertex of the parabola.

Step2: Determine the vertex of (g(x))

From the graph, the vertex of (g(x)) is ((3,7)). So (h = 3) and (k=7).

Step3: Determine the value of (a)

The parent function is (f(x)=x^{2}), and since there is no vertical stretch or compression (the parabola has the same "width" as (y = x^{2})), (a = 1).

Step4: Write the function rule for (g(x))

Substitute (a = 1), (h = 3), and (k = 7) into the vertex form (y=a(x - h)^2 + k). We get (g(x)=(x - 3)^2+7)

Answer:

(g(x)=(x - 3)^2+7)