Chapter 1.4, Problem 25E

To determine

Tocompare:The graph of given function with the graph of parent its function.

Answer to Problem 25E

The function $y={\left(x-4\right)}^3$ is obtained by shifting the parent function $y=x^3$ through a distance 4 units to the right.

Explanation of Solution

Given information:

The given function is $y={\left(x-4\right)}^3$ .

Concept used:

A parent function is the simplest function of a family of functions. The family of quadratic function $y=a{x}^2+bx+c$ has the simplest form function $y=x^2$ that is the parent function of this family of functions.

Transformations of Parent Functions:

The parent function $f\left(x\right)$ can be transformed in certain way .For, example: shift graphs up, down, left, and right.

Vertical Shifts: $f\left(x\right)+c$ moves up, $f\left(x\right)-c$ moves down.

Horizontal Shifts: $f\left(x+c\right)$ moves left, $f\left(x-c\right)$ moves right.

Graph:

The given function is

  $y={\left(x-4\right)}^3$........(1)

Let $y=f\left(x\right)$ be the parent function of function $y={\left(x-4\right)}^3$ .

Now, the parent function is movesright by extent c. Then the transformed function (given function) is

  $y=f\left(x-c\right)$........(2)

Comparing results (1) and (2), $y=x^3$ is parent function, and $y={\left(x-4\right)}^3$ is the transformed function.

The graphs of functions $y={\left(x-4\right)}^3$ and $y=x^3$ are shown in Figure-1 here.

  

Comparison:

The function $y={\left(x-4\right)}^3$ is obtained by shifting the parent function $y=x^3$ through a distance 4 units along + x axis (i.e., to the right).

Conclusion:

The function $y={\left(x-4\right)}^3$ is obtained by shifting the parent function $y=x^3$ through a distance 4 units to the right.