Chapter 1.1, Problem 33E

To determine

To graph: The given function and its parent function and then to describe the transformation.

Answer to Problem 33E

The graph of $f(x)=-{(x+3)}^2+\frac{1}{4}$ is reflection over $x-axis$ and horizontal translation left by $3units$ and vertical translation up $\frac{1}{4}$ of a unit with respect to the graph of the parent quadratic function.

Explanation of Solution

Given information: Given function is $f(x)=-{(x+3)}^2+\frac{1}{4}$ .

Graph:

The function $f$ is a quadratic function. Using a table of values to graph the given function and the parent function $g(x)=x^2$ ,

  $\begin{array}{lll}x\hfill & g(x)=x^2\hfill & f(x)=-{(x+3)}^2+\frac{1}{4}\hfill \\ -2\hfill & 4\hfill & -1+\frac{1}{4}=-\frac{3}{4}\hfill \\ -1\hfill & 1\hfill & -4+\frac{1}{4}=-\frac{15}{4}\hfill \\ 0\hfill & 0\hfill & -9+\frac{1}{4}=-\frac{35}{4}\hfill \\ 1\hfill & 1\hfill & -16+\frac{1}{4}=-\frac{63}{4}\hfill \\ 2\hfill & 4\hfill & -25+\frac{1}{4}=-\frac{99}{4}\hfill \end{array}$

So, the required graph is:

  

Interpretation:

The graph of $f$ (shown by blue colored line) is a reflection over $x-axis$ and horizontal translation left by $3units$ and vertical translation up $\frac{1}{4}$ of a unit w.r.t. the graph of the parent quadratic function $g(x)=x^2$ (shown by green colored line).

So, the graph of $f(x)=-{(x+3)}^2+\frac{1}{4}$ is a reflection over $x-axis$ and horizontal translation left by $3units$ and vertical translation up $\frac{1}{4}$ of a unit with respect to the graph of the parent quadratic function.