Chapter 1.6, Problem 67E

Solution

(a.)

The graph of the given function in the given viewing window, the prediction for how the graph of $y=\left|f\left(x\right)\right|$ will look like; and finally, the graph of $y=\left|f\left(x\right)\right|$ .

It has been determined that the graph of the given function in the given viewing window is:

  

It was predicted that for the graph of $y=\left|f\left(x\right)\right|$ will be identical to the original graph except for the parts where $y$ is negative. Those parts in the resulting graph will be a reflection of those parts in the original graph about $x$ axis, hence turning the $y$ values positive.

It has been determined that the graph of $y=\left|f\left(x\right)\right|$ is

  

Finally, it has been determined that the prediction was correct.

Given:

The function, $f\left(x\right)=x^4-5x^3+4x^2+3x+2$ and the viewing window; $\left[-5,5\right]$ by $\left[-10,10\right]$ .

Concept used:

Absolute value function turns a negative quantity into a positive one.

Calculation:

The given function is, $f\left(x\right)=x^4-5x^3+4x^2+3x+2$ and the given viewing window is $\left[-5,5\right]$ by $\left[-10,10\right]$ .

The graph of this function is given by,

  

Now, as discussed previously, the absolute value function turns a negative quantity into a positive one.

Then, it can be predicted that the graph of $y=\left|f\left(x\right)\right|$ will be identical to the above graph except for the parts where $y$ is negative. That is, the resulting graph of the parts where $y$ is negative will be a reflection of those parts in the original graph about $x$ axis, hence turning the $y$ values positive.

Now, the graph of $y=\left|f\left(x\right)\right|$ is given as,

  

It can be seen from the above graph and after a comparison with the original graph that the prediction was correct.

Conclusion:

It has been determined that the graph of the given function in the given viewing window is:

  

It was predicted that for the graph of $y=\left|f\left(x\right)\right|$ will be identical to the original graph except for the parts where $y$ is negative. Those parts in the resulting graph will be a reflection of those parts in the original graph about $x$ axis, hence turning the $y$ values positive.

It has been determined that the graph of $y=\left|f\left(x\right)\right|$ is

  

Finally, it has been determined that the prediction was correct.

(b.)

The prediction for how the graph of $y=f\left(\left|x\right|\right)$ will look like; and then, the graph of $y=f\left(\left|x\right|\right)$ .

It was predicted that the graph of $y=f\left(\left|x\right|\right)$ will be identical to the original graph for non-negative $x$ . Moreover, the reflection of this part about the $y$ axis will also be included in the resulting graph.

It has been determined that the graph of $y=f\left(\left|x\right|\right)$ is

  

Finally, it has been determined that the prediction was correct.

Given:

The function, $f\left(x\right)=x^4-5x^3+4x^2+3x+2$ and the viewing window; $\left[-5,5\right]$ by $\left[-10,10\right]$ .

Concept used:

Absolute value function turns a negative quantity into a positive one.

Calculation:

As determined previously, the graph of the original function is:

  

Now, as discussed previously, the absolute value function turns a negative quantity into a positive one.

Then, it can be predicted that the graph of $y=f\left(\left|x\right|\right)$ will be identical to the original graph for non-negative $x$ . Moreover, the reflection of this part about the $y$ axis will also be included in the resulting graph.

Now, the graph of $y=f\left(\left|x\right|\right)$ is given as,

  

It can be seen from the above graph and after a comparison with the original graph that the prediction was correct.

Conclusion:

It was predicted that the graph of $y=f\left(\left|x\right|\right)$ will be identical to the original graph for non-negative $x$ . Moreover, the reflection of this part about the $y$ axis will also be included in the resulting graph.

It has been determined that the graph of $y=f\left(\left|x\right|\right)$ is:

  

Finally, it has been determined that the prediction was correct.

(c.)

The graph of $y=\left|g\left(x\right)\right|$ .

It has been determined that the graph of $y=\left|g\left(x\right)\right|$ is given as,

  

Given:

The graph of $y=g\left(x\right)$, as follows:

  

Concept used:

Absolute value function turns a negative quantity into a positive one.

Calculation:

It is given that the graph of $y=g\left(x\right)$ is

  

Now, as discussed previously, the absolute value function turns a negative quantity into a positive one.

Then, it can be predicted that the graph of $y=\left|g\left(x\right)\right|$ will be identical to the above graph except for the parts where $y$ is negative. That is, the resulting graph of the parts where $y$ is negative will be a reflection of those parts in the original graph about $x$ axis, hence turning the $y$ values positive.

Now, the graph of $y=\left|g\left(x\right)\right|$ is given as,

  

Conclusion:

It has been determined that the graph of $y=\left|g\left(x\right)\right|$ is given as,

  

(d.)

The graph of $y=g\left(\left|x\right|\right)$ .

It has been determined that the graph of $y=g\left(\left|x\right|\right)$ is given as,

  

Given:

The graph of $y=g\left(x\right)$, as follows:

  

Concept used:

Absolute value function turns a negative quantity into a positive one.

Calculation:

It is given that the graph of $y=g\left(x\right)$ is

  

Now, as discussed previously, the absolute value function turns a negative quantity into a positive one.

Then, it can be predicted that the graph of $y=g\left(\left|x\right|\right)$ will be identical to the original graph for non-negative $x$ . Moreover, the reflection of this part about the $y$ axis will also be included in the resulting graph.

Now, the graph of $y=g\left(\left|x\right|\right)$ is given as,

  

Conclusion:

It has been determined that the graph of $y=g\left(\left|x\right|\right)$ is given as,