Problem 144 - 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 \(f(x)=x^{3}\) 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: Understand the Transformations

Investigate the suggested transformations: shift three units to the right, reflect about the \(x\)-axis and then move down four units.

Step 2: Apply Horizontal Shift

First, apply a horizontal shift of three units to the right on function \(f\). During a horizontal shift, the value of \(x\) changes while \(y\) remains constant. The shift to the right is represented by replacing \(x\) with \(x-c\) (where \(c\) is the units of shift), in the function. In this case, \(x\) is replaced with \(x-3\). Therefore, \(f(x) = x^{3}\) becomes: \((x-3)^{3}\).

Step 3: Apply Reflection

Second, apply reflection about the \(x\)-axis. A reflection over the \(x\)-axis changes the sign of \(y\). Therefore, \(f(x)=(x-3)^{3}\) becomes: \[-(x-3)^{3}\]

Step 4: Apply Vertical Shift

Finally, apply a downward vertical shift of four units. A vertical shift changes the value of \(y\) while \(x\) remains constant. A downward shift is represented by subtracting the value from the function. Therefore, \(- (x-3)^{3}\) becomes: \[-(x-3)^{3} - 4\]. Now, the function matches with function \(g(x) = -(x-3)^{3} - 4\)

Step 5: Conclusion

Conclude that the statement is true since applying the transformations on function \(f\), results to function \(g\).