Problem 144 - 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 \(f(x)=x^{3}\) and \(g(x)=-(x-3)^{3}-4,\) then the graph of \(g\) can be obtained from the graph of \(f\) by moving \(f\) three units to the right, reflecting about the \(x\) -axis, and then moving the resulting graph down four units.

Step-by-Step Solution

Answer

The statement is True

Step 1: Identify the Characteristics

Notice the properties of the compared functions \(f(x)=x^{3}\) and \(g(x)=-(x-3)^{3}-4\). We are told that g is derived from f by moving right three units, reflecting about the x-axis, and moving down four units.

Step 2: Verify the Transformations

Now, examine the components of \(g(x)=-(x-3)^{3}-4\). The \(x-3\) in the equation indicates a shift to the right by 3 units. The negative sign in front of the parentheses implies a reflection about the x-axis. The -4 signifies a downward shift of 4 units. All the transformations align with the question statement.

Step 3: Final Statement

After verifying that every transformation aligns with the question, we can conclude that the statement is True.