Problem 141 - College Algebra

Question

Determine whether each 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 \(f\) three units to the right and three units upward.

Step-by-Step Solution

Answer

The statement is false. The correct statement is 'If \(f(x)=|x|\) and \(g(x)=|x+3|+3,\) then the graph of \(g\) is a translation of the graph of \(f\) three units to the left and three units upward.

Step 1: Analyze the given functions

Looking at the functions, \(f(x)\) is simply the absolute value of \(x\), graphed as a V shape centered on the y-axis. \(g(x)\) on the other hand, contains adjustments inside and outside the absolute value.

Step 2: Determine effects of adjustments on translation of graph

From the function \(g(x)\), the \(+3\) inside the absolute value will shift the graph of \(|x|\) three units to the left, not to the right - transformation rules indicate that a positive value added to \(x\) inside a function \(f(x)\) move the graph to the left, and a negative value moves it to the right. The \(+3\) outside the absolute value indeed shifts the graph of \(|x|\) three units up.

Step 3: Conclude and correct the given statement

Based on the analysis, the given statement is incorrect. The function \(g(x)=|x+3|+3\) shifts the graph of \(f(x)=|x|\) three units to the left, not to the right, and three units up.