Chapter 1.4, Problem 29E

To determine

To identify:The parent function and the transformation shown in the graph and write an equation for the function represented by the graph.

Answer to Problem 29E

The parent function of function is $f\left(x\right)=x^3$ and equation for the function represented by the given graph is $g\left(x\right)={\left(x-2\right)}^3$ .

Explanation of Solution

Given information:

The given function is shown in Figure-1.

Concept used:

A parent function is the simplest function of a family of functions represents the definition (or shape) of the family of functions. For example, the family of quadratic functions has the general form $g\left(x\right)=a{x}^2+bx+c$ . The simplest function or parent function of this family is $f\left(x\right)=x^2$ .

Identification:

The shape of given graph (Figure-1) appears similar to thegraph of a cubicfunction $f\left(x\right)=x^3$ whose graph is shown in Figure-2.

If we shift the graph of function $f\left(x\right)=x^3$ to the right along +x axis by 2 unit, then we get the graph of given function shown in Figure-1.

The equation of function obtained by shifting the graph of function $f\left(x\right)$ to the right along +x axis is given by $f\left(x-2\right)$ .

Thus, equation the transformed function say $g\left(x\right)$ is $g\left(x\right)=f\left(x-2\right)={\left(x-2\right)}^3$ .

The graphs functions $f\left(x\right)=x^3$ and $g\left(x\right)={\left(x-2\right)}^3$ are shown in Figure-2.

  

Thus, the parent function of the function whose graph is shown in given figure (Figure-1) is $f\left(x\right)=x^3$ and equation for the function represented by the given graph is $g\left(x\right)={\left(x-2\right)}^3$ .

Conclusion:

The parent function of function is $f\left(x\right)=x^3$ and equation for the function represented by the given graph is $g\left(x\right)={\left(x-2\right)}^3$ .