Q. 36 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 31–36, graph each function using the techniques of shifting, compressing or stretching, and reflections. Identify any intercepts on the graph. State the domain and, based on the graph, find the range. 

$g\left(x\right)=-2{(x+2)}^3-8$

Step-by-Step Solution

Answer

The domain of the function is the set of all real numbers.

The intercepts are $(-3.6,0) and (0,-24).$

The range of the function is the set of all real numbers.

The graph of the function is

Step 1. Given Information

The given function is $g\left(x\right)=-2{(x+2)}^3-8.$

We have to graph the function using the techniques of shifting, compressing, or stretching, and reflections then find the domain, intercepts, and range.

Step 2. Sketching the graph of the function

The basic function of the given function is $g\left(x\right)=x^3.$

To obtain the graph of g(x) from the graph of basic function, shift the graph horizontally to the left by $2$ units then multiply $2$ by each y-coordinate on the graph of basic function. Then reflect the graph on the x-axis and then shift vertically down by $8$ units.

The graph of g(x) will be identical to the graph of basic function, except that it is shifted horizontally to the left by $2$ units and then multiplied by $2$ and reflected on the x-axis then shift vertically down by $8$ units.

Step 3. Graph of the function

Step 4. Finding the domain of the function

The domain of the function is the set of all real numbers because the function $g\left(x\right)=-2{(x+2)}^3-8$ can be performed on any real numbers.

Step 5. Identifying the intercepts

The points where a line crosses the x-axis and the y–axis are called the intercepts.

Thus, the graph touches the axis at $(-3.6,0) and (0,-24).$

Step 6. Finding the range

From the graph, we conclude that y-coordinates are between $-\infty and \infty .$

Thus, the range is the set of all real numbers.