Question

determine whether the following statement is true or false. if the statement is false, make the necessary change(s) to produce a true statement. if ( f(x) = |x| ) and ( g(x) = |x + 3| + 3 ), then the graph of ( g ) is a translation of the graph of ( f ) 3 units to the right and 3 units upward. choose the correct answer below. a. the statement is false, the graph of ( f ) is shifted 3 units to the right and 3 units downward. b. the statement is false, the graph of ( f ) is shifted 3 units to the left and 3 units upward. c. the statement is false, the graph of ( f ) is shifted 3 units to the left and 3 units downward. d. the statement is true.

Explanation:

Step1: Recall horizontal shift rules

For a function $f(x)$, $f(x + h)$ shifts the graph left by $h$ units when $h>0$.

Step2: Recall vertical shift rules

For a function $f(x)$, $f(x) + k$ shifts the graph up by $k$ units when $k>0$.

Step3: Analyze $g(x)$ relative to $f(x)$

Given $f(x)=|x|$ and $g(x)=|x+3|+3=f(x+3)+3$, this means a left shift of 3 units and an upward shift of 3 units. The original statement claims a right shift, so it is false.

Answer:

B. The statement is false, the graph of f is shifted 3 units to the left and 3 units upward.