Chapter 12, Problem 77CLR

To determine

To graph: The function $f\left(x\right)=4-2x-x^2$ .

Answer to Problem 77CLR

The graph is shown in Figure 1.

Explanation of Solution

Calculation:

Any quadratic equation of the form $a{x}^2+bx+c=0$ represents a parabola with the x -coordinate of the vertex given by $x=\frac{-b}{2a}$ .

So, for the function $f\left(x\right)=4-2x-x^2$ ,

  $\begin{array}{c}a=-1\\ b=-2\\ c=4\end{array}$

The x -coordinate of the vertex is,

  $\begin{array}{c}x=\frac{-b}{2a}\\ =\frac{-\left(-2\right)}{2\left(-1\right)}\\ =-1\end{array}$

Substitute -1 for x in the expression $4-2x-x^2$ to find the y-coordinate of the parabola.

  $\begin{array}{c}4-2x-x^2=4-2\left(-1\right)-{\left(-1\right)}^2\\ =4+2-1\\ =5\end{array}$

With the vertex as $\left(-1,5\right)$ , the parabola $f\left(x\right)=4-2x-x^2$

is plotted in a graphing utility as shown in the figure below:

  

Figure 1