Problem 141 - College Algebra Essentials

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 should be: 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: Understanding the Transformation

The given functions are \(f(x)=|x|\) and \(g(x)=|x+3|+3\). Now, adding a constant to the function is a vertical transformation. This means it will move the graph upwards or downwards. In this case \(3\) is added, so the graph will move \(3\) units upwards.

Step 2: Analyze the Effect of \(|x+3|\)

The expression \(|x+3|\) is a horizontal shift of the original function \(|x|\). If a constant is added or subtracted inside the absolute value, the function will shift right if the constant is subtracted and left if the constant is added. Here the constant \(3\) is added, so the graph should shift \(3\) units to the left, not right as the statement suggests.

Step 3: Determine Truth Value of the Statement

Since the \(3\) added inside the absolute value causes a shift to the left, not right, the initial statement is false.

Step 4: Correct the Statement

To make the statement true, the direction of the horizontal shift must be corrected. So, the correct statement would be: 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.