Q. 64 - Precalculus Enhanced with Graphing Utilities

Question

In Problems $39-68$, graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function (for example, $y=x^2$) and show all stages. Be sure to show at least three key points. Find the domain and the range of each function.Verify your results using a graphing utility.

$f\left(x\right)=-4\sqrt{x-1}$

Step-by-Step Solution

Answer

The graph of the function $f\left(x\right)=-4\sqrt{x-1}$ is:

The domain of the function is $\{x\in \mathrm{ℝ}:x\ge 1\}$ and the range is the set of all non-positive real numbers.

Step 1. Given

The function $f\left(x\right)=-4\sqrt{x-1}$

To graph the function and to find its domain and range.

Step 2. Graph the basic function

Graph the basic function $y=\sqrt{x}$

Step 3. Replace x with x - 1

Replace $y=\sqrt{x}$ by

               $y=\sqrt{x-1}$ so that the graph shifted horizontally right to $1$ unit.

Step 4. Multiply the right side by - 4

Multiply the graph $y=\sqrt{x-1}$ by $-4$ so that he graph stretched by the factor of $4$ and reflected about $x-$axis.

Step 5. Find domain and range

The domain of the function is $\{x\in \mathrm{ℝ}:x\ge 1\}$ and the range is the set of all non-positive real numbers.

Step 6. Verify the graph

Verify the graph by using graphing utility.

The graph of the function using graphing utility is: