Chapter 1.2, Problem 32E

Solution

To identify: graph the function and intervals on which the function is increasing,

decreasing, or constant.

The function is

Decreases: $(-\infty ,-2]$

Increases: $[-2,\infty )$ .

Given information: 

The function $h\left(x\right)=0.5{\left(x+2\right)}^2-1$ .

Calculation:

Graph of

  $h\left(x\right)=0.5{\left(x+2\right)}^2-1$

Let $y=h\left(x\right)$ .

Putting the value of $x=1$ then $y=3.5$ .

And $x=-1$ then $y=0.5$

And $x=-2$ then $y=-1$

And $x=2$ then $y=7$

And $x=\infty$ then $y=\infty$ .

Plot the graph:

  

Therefore, the function decreases in the interval $(-\infty ,-2]$ and increases in the interval $[-2,\infty )$ .