Chapter 2.2, Problem 18E

(a)

To determine

To sketch : The graph of $y=x^n$ and each transformation.

Explanation of Solution

Given: The given function is $y=x^6$ and transformation is $f\left(x\right)=-\frac{1}{8}{x}^6$ .

The graph of $y=x^6$ is shown in figure below.

  

Figure (1)

The leading coefficient test suggests that as the degree is even and the leading coefficient is positive, so both side of the graph will rise up.

Now consider the transformation $f\left(x\right)=-\frac{1}{8}{x}^6$ .

The graph of $y=x^6$ when given a vertical shrink by a factor of $\frac{1}{8}$ followed by reflection about the $x$ axis produces the graph of the $y=-\frac{1}{8}x$ .

The graph of $y=-\frac{1}{8}x$ is shown in figure below.

  

Figure (2)

Therefore, the graph of the function $y=x^6$ is shown in Figure (1) and the transformation $f\left(x\right)=-\frac{1}{8}x$ is shown in Figure (2).

(b)

To determine

To sketch : The graph transformation $f\left(x\right)={\left(x+2\right)}^6-4$

Explanation of Solution

Given: The given transformation is $f\left(x\right)={\left(x+2\right)}^6-4$ .

Consider the transformation $f\left(x\right)={\left(x+2\right)}^6-4$ .

The graph of $y=x^6$ when shifted two unit to the left followed by $4$ units in the downward direction produces $y={\left(x+2\right)}^6-4$

The graph of $y={\left(x+2\right)}^6-4$ is shown in figure below.

  

Figure (3)

Therefore, the graph of the transformation $f\left(x\right)={\left(x+2\right)}^6-4$ is shown in Figure (3).

(c)

To determine

To sketch : The graph transformation $f\left(x\right)={\left(x\right)}^6-5$

Explanation of Solution

Given: The given transformation is $f\left(x\right)={\left(x\right)}^6-5$ .

Consider the transformation $f\left(x\right)={\left(x\right)}^6-5$ .

The graph of $y=x^6$ when shifted by five units in the downward direction produces $y={\left(x\right)}^6-5$

The graph of $f\left(x\right)={\left(x\right)}^6-5$ is shown in figure below.

  

Figure (4)

Therefore, the graph of the transformation $f\left(x\right)={\left(x\right)}^6-5$ is shown in Figure (4).

(d)

To determine

To sketch : The graph transformation $f\left(x\right)=\frac{-1}{4}{x}^6+1$

Explanation of Solution

Given: The given transformation is $f\left(x\right)=\frac{-1}{4}{x}^6+1$ .

Consider the transformation $f\left(x\right)=\frac{-1}{4}{x}^6+1$ .

The graph of $y=x^6$ when given a vertical shrink of $\frac{1}{4}$ units followed by a reflection about the $x$ axis then shift of $1$ unit in the upward direction produces $y=\frac{-1}{4}{x}^6+1$

The graph of $f\left(x\right)=\frac{-1}{4}{x}^6+1$ is shown in figure below.

  

Figure (5)

Therefore, the graph of the transformation $f\left(x\right)=\frac{-1}{4}{x}^6+1$ is shown in Figure (5).

(e)

To determine

To sketch : The graph transformation $f\left(x\right)=\frac{1}{4096}{x}^6-2$

Explanation of Solution

Given: The given transformation is $f\left(x\right)=\frac{1}{4096}{x}^6-2$ .

Consider the transformation $f\left(x\right)=\frac{1}{4096}{x}^6-2$ .

The graph of $y=x^6$ when given a vertical shrink of $\frac{1}{4096}$ units and then shift of $2$ unit in the downward direction produces $y=\frac{1}{4096}{x}^6-2$

The graph of $f\left(x\right)=\frac{1}{4096}{x}^6-2$ is shown in figure below.

  

Figure (6)

Therefore, the graph of the transformation $f\left(x\right)=\frac{1}{4096}{x}^6-2$ is shown in Figure (6).

(f)

To determine

To sketch : The graph transformation $f\left(x\right)={\left(2x\right)}^6-1$

Explanation of Solution

Given: The given transformation is $f\left(x\right)={\left(2x\right)}^6-1$ .

Consider the transformation $f\left(x\right)={\left(2x\right)}^6-1$ .

The transformation can be re written as $f\left(x\right)=64x^6-1$

The graph of $y=x^6$ when given a vertical stretch of $64$ units and then shift of $1$ unit in the downward direction produces $y={\left(2x\right)}^6-1$

The graph of $f\left(x\right)={\left(2x\right)}^6-1$ is shown in figure below.

  

Figure (7)

Therefore, the graph of the transformation $f\left(x\right)={\left(2x\right)}^6-1$ is shown in Figure (7).