Chapter 1.4, Problem 31E

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 31E

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

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 type of thegraph of function $f\left(x\right)=x^2$ whose graph is shown in Figure-2.

Thus, the parent function of the given graph is $f\left(x\right)=x^2$ .

If we shift the graph of function $f\left(x\right)=x^2$ to the up along - y axis by 3 units, 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 down along -y axis by unit cis given by $f\left(x\right)-c$ .

Hence, the equation the transformed function say $g\left(x\right)$ is $g\left(x\right)=f\left(x\right)-3$ or $g\left(x\right)=x^2-3$

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

  

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

Conclusion:

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