Chapter 2.1, Problem 34E

To determine

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

Answer to Problem 34E

  $f(x)={(x+6)}^2+3$ , with a horizontal shrink by a factor $\frac{1}{2}$ and a reflection in the $x-\text{axis}$ , followed by a translation $1$ unit down is $g(x)=-{(2x+6)}^2+2$ .

Vertex of the given function is $(-3,2)$ .

Explanation of Solution

Given information: $f(x)={(x+6)}^2+3$ with a horizontal shrink by a factor $\frac{1}{2}$ and a reflection in the $x-\text{axis}$ , followed by a translation $1$ units down.

Describing the transformation:

Horizontal shrink occurs when $a>1$ , here $a=2$ .

Translation of $1$ units down occurs when $k$ is on the negative side.

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

  $g(2x)=-{(2x+6)}^2+3-1$

  $y=-{(2x+6)}^2+2$

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=-{(2x+6)}^2+2$$ \text{By}\text{using}{(a+b)}^2=a^2+b^2+2ab,$y=-4x^2-24x-36+2\\ y=-4x^2-24x-34\\ a=-4,b=-24,c=-34.\end{array}$

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

  $\begin{array}{l}x=-\frac{(-24)}{2\times (-4)}\\ x=-\frac{(-24)}{(-8)}\\ x=-3\end{array}$

Substitute $x=-3$ in $y=-4x^2-24x-34$ ,

  $\begin{array}{l}y=-4x^2-24x-34\\ $ y=-4{(-3)}^2-24(-3)-34$y=-4(9)-24(-3)-34\\ y=-36+72-34\\ y=2\end{array}$

The vertex of the given function is $(-3,2)$ .