Chapter 1.7, Problem 55E

(a)

To determine

To find: The function for the given graph.

Answer to Problem 55E

Therequired function is $g\left(x\right)=-3x^2$ .

Explanation of Solution

Given:

The given function is $f\left(x\right)=x^2$ .

The given graph is shown in Figure 1

  

Figure 1

Calculation:

The parent function for the above graph is,

  $f\left(x\right)=x^2$

The equation of the new function is $g\left(x\right)$ that is obtained by function $f\left(x\right)=x^2$ by reflecting the function $f\left(x\right)=x^2$ along the $x$ axis as,

  $g\left(x\right)=-x^2$

Then stretch the above function vertically by a factor of $k$ ,

  $g\left(x\right)=-k{x}^2$

The graph for $g\left(x\right)$ passes through the points $\left(1,-3\right)$ ,

  $\begin{array}{l}-k{\left(1\right)}^2=-3\\ k=3\end{array}$

Then,

  $g\left(x\right)=-3x^2$

(b)

To determine

To find: The function for the given graph.

Answer to Problem 55E

The required function is $g\left(x\right)=4x^2+3$ .

Explanation of Solution

Given:

The given function is $f\left(x\right)=x^2$ .

The given graph is shown in Figure 2

  

Figure 2

Calculation:

The parent function for the above graph is,

  $f\left(x\right)=x^2$

The equation of the new function is $g\left(x\right)$ that is obtained by function $f\left(x\right)=x^2$ asm

  $g\left(x\right)=x^2$

Then stretch the above function vertically by a factor of $k$ ,

  $g\left(x\right)=k{x}^2$

Then shift the graph vertically three units upward as,

  $g\left(x\right)=k{x}^2+3$

The graph for $g\left(x\right)$ passes through the points $\left(1,7\right)$ ,

  $\begin{array}{l}k{\left(1\right)}^2+3=7\\ k=4\end{array}$

Then,

  $g\left(x\right)=4x^2+3$