Chapter 1.2, Problem 2Q

To determine

To identify: The function family to which $g$ belongs and then to compare the graph of the function to the graph of its parent function.

Answer to Problem 2Q

Parent function is $f(x)=x^2$ , $g$ is translated $1$ unit to the left and is vertically stretched by a factor of $2$ .

Explanation of Solution

Given information: Function is $g(x)=2{(x+1)}^2$ .

Calculation:

  $\begin{array}{l}$ g(x)=2{(x+1)}^2$\\ =2(x^2+1+2x)\\ \\ =2x^2+2x+2\end{array}$

  $g$ belongs to family of quadratic functions,

Parent function is $f(x)=x^2$ .

  

Red line represents the given function whereas blue line represents parent function.

Given function is translated $1$ unit to the left and is vertically stretched by a factor of $2$ with respect to the parent function.

Domain of parent function: All real numbers

Range of parent function: $y\ge 0$

Domain of given function: All real numbers

Range of given function: $y\ge 0$

Thus, parent function is $f(x)=x^2$ , $g$ is translated $1$ unit to the left and is vertically stretched by a factor of $2$ .