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

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

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

Answer

Explanation:

Step1: Identify the vertex

The vertex of the absolute - value function (y = a|x - h|+k) is ((h,k)). From the graph, the vertex of (g(x)) is ((4, - 2)), so (h = 4) and (k=-2).

Step2: Find the value of (a)

We know that the parent function is (y = |x|). Let's take a non - vertex point, say ((0,-8)). Substitute (x = 0), (y=-8), (h = 4), and (k=-2) into (y=a|x - h|+k). We get (-8=a|0 - 4|-2). First, simplify the absolute - value: (-8=a| - 4|-2), and (| - 4| = 4), so the equation becomes (-8 = 4a-2). Add 2 to both sides: (-8 + 2=4a-2 + 2), which gives (-6 = 4a). Solve for (a): (a=-\frac{3}{2}).

Answer:

(g(x)=-\frac{3}{2}|x - 4|-2)