Chapter 2, Problem 64RE

To determine

To sketch: the graph of the function. Also state the domain, intercepts and range of the graph.

Answer to Problem 64RE

The range is $\left\{y|y\ge -3\right\}$ .

Explanation of Solution

Given information:

Given function

  $h\left(x\right)={\left(x+2\right)}^2-3$

Calculation:

Let us consider the following function

  $h\left(x\right)={\left(x+2\right)}^2-3$

The function $h\left(x\right)={\left(x+2\right)}^2-3$ has the same shape as the graph of $f\left(x\right)=x^2$ .

The graph of $h\left(x\right)={\left(x+2\right)}^2-3$ can be obtained from the graph of $f\left(x\right)=x^2$ by applying the following steps.

1. Shift horizontally left 2 units.
2. Shift vertically 3 units downwards.

The graph $h\left(x\right)={\left(x+2\right)}^2-3$ is plotted below using Maple.

  

The function $h\left(x\right)={\left(x+2\right)}^2-3$ has no restriction on the values of $x$ can take. Hence the domain of the function $h\left(x\right)={\left(x+2\right)}^2-3$ is the set of all real numbers. From the graph we see $h\left(x\right)$ is always greater than $-3$ . Hence the range of the function $h\left(x\right)$ is $\left\{y|y\ge -3\right\}$ .