Question

function g can be thought of as a translated (shifted) version of ( f(x) = |x| ).
what is the equation for ( g(x) )?
choose 1 answer:
a ( g(x) = |x + 7| + 4 )
b ( g(x) = |x - 4| + 7 )
c ( g(x) = |x + 4| + 7 )
d ( g(x) = |x - 7| + 4 )

Explanation:

Step1: Recall transformation of absolute function

The parent function is \( f(x) = |x| \), with vertex at \((0,0)\). For a transformed absolute function \( g(x)=|x - h|+k \), the vertex is \((h,k)\), where \( h \) is horizontal shift (\( h>0 \) right, \( h<0 \) left) and \( k \) is vertical shift (\( k>0 \) up, \( k<0 \) down).

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

From the graph, the vertex of \( g(x) \) (the corner of the V - shape) is at \((7,4)\). So \( h = 7 \) (since vertex x - coordinate is 7, \( x - h=x - 7 \)) and \( k = 4 \) (y - coordinate of vertex is 4).

Step3: Form the equation of \( g(x) \)

Substitute \( h = 7 \) and \( k = 4 \) into \( g(x)=|x - h|+k \). We get \( g(x)=|x - 7|+4 \), which corresponds to option D.

Answer:

D. \( g(x)=|x - 7|+4 \)