Chapter 2.7, Problem 45PPSE

Solution

a.

To Graph:

  $f\left(x\right)=-\frac{1}{2}\left|x-3\right|$ and $g\left(x\right)=\left|-\frac{1}{2}\left(x-3\right)\right|$

Given:

  $f\left(x\right)=-\frac{1}{2}\left|x-3\right|$ and $g\left(x\right)=\left|-\frac{1}{2}\left(x-3\right)\right|$

Concepts Used:

Plotting linear equations like $y=ax+b$ by determining points on its graph and joining them.

The graph of equation $y=\left|x\right|$ has its lowest point (the vertex) at the origin. Towards the right of origin ( $x>0$ ) it is equivalent to the graph of $y=x$ whereas it is equivalent to the graph of $y=-x$ towards the left of the origin ( $x<0$ ).

The graph of $y=f\left(x-a\right);a>0$ is the graph of $y=f\left(x\right)$ shifted to the right by $a$ units.

The graph of $y=af\left(x\right);a>0$ is the graph of $y=f\left(x\right)$ stretched along the $y$ axis by a factor of $a$ .

The graph of $y=-f\left(x\right)$ is the graph of $y=f\left(x\right)$ reflected about the $x$ axis.

Graph

  

Conclusion:

The graph of $f\left(x\right)=-\frac{1}{2}\left|x-3\right|$ and $g\left(x\right)=\left|-\frac{1}{2}\left(x-3\right)\right|$ are shown in the plot above.

b.

To Describe:

The similarities and differences between the two graphs $y=f\left(x\right)$ and $y=g\left(x\right)$ .

Both the graphs are similar to the graph of an absolute value function having the point $\left(0,3\right)$ as the vertex.

They are reflected versions of each other about the $x$ axis. While $y=g\left(x\right)$ lies in the upper half plane, $y=f\left(x\right)$ lies in the lower half plane.

Given:

  $f\left(x\right)=-\frac{1}{2}\left|x-3\right|$ and $g\left(x\right)=\left|-\frac{1}{2}\left(x-3\right)\right|$

Known from the previous part:

The graphs of $y=f\left(x\right)$ and $y=g\left(x\right)$ .

Concepts Used:

Observing and interpreting graphs of functions.

Description:

Both the graphs are similar to the graph of an absolute value function having the point $\left(0,3\right)$ as the vertex.

They are reflected versions of each other about the $x$ axis. While $y=g\left(x\right)$ lies in the upper half plane, $y=f\left(x\right)$ lies in the lower half plane.