Chapter 2.1, Problem 31E

To determine

To find: Describe the transformations of graph of $f$ and identify the vertex.

Answer to Problem 31E

  $f(x)=x^2$ , with a vertical stretch by a factor $4$ and a reflection in the $x-\text{axis}$ , followed by a translation $2$ units up is $g(x)=-4x^2+2$ .

Vertex of the given function is $(0,2)$ .

Explanation of Solution

Given information: $f(x)=x^2$ with a vertical stretch by a factor $4$ and a reflection in the $x-\text{axis}$ , followed by a translation $2$ units up.

Describing the transformation:

Vertical stretch occurs when $a>1$ , here $a=4$ .

Translation of $2$ units up occurs when $k$ is on the positive side.

Reflection in the $x-\text{axis}$ is $y=-x^2$ .

  $g(x)=-4x^2+2$

Therefore, the above function is the rule of $g$ described by the transformations of the graph $f$ .

Identifying the vertex:

General form of a quadratic function is $y=a{x}^2+bx+c$ .

  $\begin{array}{l}y=-4x^2+2\\ a=-4,b=0,c=2.\end{array}$

The $x-$ coordinate of the vertex of the given parabola is $x=-\frac{b}{2a}$ ,

  $\begin{array}{l}x=-\frac{0}{2\times (-4)}\\ x=-\frac{0}{(-8)}\\ x=0\end{array}$

Substitute $x=0$ in $g(x)=-4x^2+2$ ,

  $\begin{array}{l}$ y=-4{(0)}^2+2$y=0+2\\ y=2\end{array}$

The vertex of the given function is $(0,2)$ .