Chapter 2.5, Problem 58AYU

In Problems 39-62, graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Stan with the graph of the basic function (for example, $y\text{ = }{x}^2$) and show all stages. Be sure to show at least three key points. Find the domain and the range of each function.

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

To determine

To graph: The function $f\left(x\right)=-4\sqrt{x -1}$, 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.

Answer to Problem 58AYU

Domain of the function $f\left(x\right)=-4\sqrt{x -1}$ is $\left[1,\infty \right)$.

Range of the function $f\left(x\right)=-4\sqrt{x -1}$ is $\left[0,-\infty \right)$.

Explanation of Solution

Given:

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

Graph:

Now use the following steps to obtain the graph of $f\left(x\right)$.

Step 1: The function $f\left(x\right)=\sqrt{x}$ is the square root function.

$f\left(x\right)=\sqrt{x}$     square root function

Step 2: To obtain the graph of $f\left(x\right)=-\sqrt{x}$, multiply by $-1$ in the graph of $f\left(x\right)=\sqrt{x}$, that it is reflection about the $x\text{-axis}$.

$f\left(x\right)=-\sqrt{x}$     multiply by $-1$, reflection about $x\text{-axis}$

Step 3: To obtain the graph of $f\left(x\right)=-\sqrt{x -1}$, replace $x$ by $x-1$ from each $y\text{-coordinate}$ on the graph of $f\left(x\right)=-\sqrt{x}$, that it is shifted right 1 unit.

$f\left(x\right)=-\sqrt{x -1}$     replace $x$ by $x-1$; Horizontal shift right 1 unit.

Step 4: To obtain the graph of $f\left(x\right)=-4\sqrt{x -1}$, multiply each $y\text{-coordinate}$ of the graph of $f\left(x\right)=-\sqrt{x -1}$, that it is vertically stretched by the factor of 4.

$f\left(x\right)=-4\sqrt{x -1}$     multiply by 4, vertically stretched by a factor of 4

Interpretation:

Domain of the function $f\left(x\right)=-4\sqrt{x -1}$ is $\left[1,\infty \right)$.

Range of the function $f\left(x\right)=-4\sqrt{x -1}$ is $\left[0,-\infty \right)$.