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

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

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

Answer

Answer:

(g(x)=|x| + 6)

Explanation:

Step1: Recall the vertex form of absolute - value function

The vertex form of an absolute - value function is (y=a|x - h|+k), where ((h,k)) is the vertex of the graph.

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

For the function (f(x)=|x|), its vertex is ((0,0)). For the function (g(x)), from the graph, the vertex is ((0,6)), so (h = 0) and (k=6).

Step3: Determine the value of (a)

Take a point on (g(x)), say when (x = 1), (y=7). Substitute (x = 1), (y = 7), (h = 0), (k = 6) into (y=a|x - h|+k). We get (7=a|1 - 0|+6). [ \begin{align*} 7&=a\times1+6\ a&=7 - 6\ a&=1 \end{align*} ]

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

Substitute (a = 1), (h = 0), (k = 6) into (y=a|x - h|+k). We have (g(x)=1\times|x-0|+6=|x| + 6).