Chapter 3.8, Problem 65PFA

a.

To determine

To determine whether the function $g(x)=-\frac{1}{2}\left|x\right|+4$ has a minimum point or not.

Answer to Problem 65PFA

False

Explanation of Solution

Given:

Function: $g(x)=-\frac{1}{2}\left|x\right|+4$

Concept used:

The minimum value from the critical points of the function is the minimum point.

Calculation:

Consider $g(x)=-\frac{1}{2}\left|x\right|+4$ ,

Min of $-\left|x\right|$ does not exist.

  $\Rightarrow$ Min of $g(x)=-\frac{1}{2}\left|x\right|+4$ does not exist.

Conclusion:

Therefore, the given function does not have a minimum point.

Therefore, the given statement is false.

b.

To determine

To determine the $x$ -intercepts of the function $g(x)=-\frac{1}{2}\left|x\right|+4$ .

Answer to Problem 65PFA

True

Explanation of Solution

Given:

Function: $g(x)=-\frac{1}{2}\left|x\right|+4$

Concept used:

The zero of the function is the $x$ -intercept of that function.

Calculation:

Consider $g(x)=-\frac{1}{2}\left|x\right|+4$ .

To find the $x$ -intercepts of $g(x)$ , put $g(x)=0$

  $\begin{array}{l}\Rightarrow -\frac{1}{2}\left|x\right|+4=0\\ \Rightarrow -\frac{1}{2}\left|x\right|=-4\\ \Rightarrow \frac{1}{2}\left|x\right|=4\\ \Rightarrow \left|x\right|=8\\ \Rightarrow x=\pm 8\end{array}$

Therefore, there are two $x$ -intercepts: $\left(-8,0\right)$ and $\left(8,0\right)$ .

Conclusion:

Therefore, the given function has two $x$ -intercepts.

Therefore, the given statement is true.

c.

To determine

To determine the vertex of the function $g(x)=-\frac{1}{2}\left|x\right|+4$ .

Answer to Problem 65PFA

True

Explanation of Solution

Given:

Function: $g(x)=-\frac{1}{2}\left|x\right|+4$

Concept used:

The general form of an absolute value function is $f(x)=a\left|x-h\right|+k$ .

The vertex of the function is $\left(h,k\right)$ .

The variables $h$ and $k$ tell how far the graph shifts horizontally and vertically.

Calculation:

Consider $g(x)=-\frac{1}{2}\left|x\right|+4$

Compare with the general form:

  $g(x)=a\left|x-h\right|+k$

Here, the value of $a=-\frac{1}{2}$ , so the graph opens downwards with a slope of $-\frac{1}{2}$ .

The value of $h=0$ and the value of $k=4$ , so the vertex of the graph is $\left(0,4\right)$ .

The value of $h=0$ represents the graph is translated horizontally by 0 units and the value of $k=4$ represents that the graph is translated up by 4 units.

Conclusion:

Therefore, the vertex of the graph is $\left(0,4\right)$ .

Therefore, the given statement is true.

d.

To determine

To determine whether the graph of the function $g(x)=-\frac{1}{2}\left|x\right|+4$ is symmetric about $x$ -axis or not.

Answer to Problem 65PFA

True

Explanation of Solution

Given:

Function: $g(x)=-\frac{1}{2}\left|x\right|+4$

Concept used:

The graph of function is said to be symmetric about $x$ -axis if $\left(x,f(x)\right)$ is on the graph, then so is $\left(x,-f(x)\right)$ .

Calculation:

Consider the function $g(x)=-\frac{1}{2}\left|x\right|+4$ .

The graph will be symmetric about $x$ -axis if $g(-x)=g(x)$

  $\Rightarrow g(-x)=-\frac{1}{2}\left|-x\right|+4$

  $\begin{array}{l}=-\frac{1}{2}\left|x\right|+4\\ =g(x)\end{array}$

  $\Rightarrow g(-x)=g(x)$

Therefore, the graph is symmetric about $x$ -axis.

Conclusion:

Therefore, the graph is symmetric about $x$ -axis.

Therefore, the given statement is true.

e.

To determine

To determine the translationin the graph of $f(x)=\left|x\right|$ to get the graph of the function $g(x)=-\frac{1}{2}\left|x\right|+4$ .

Answer to Problem 65PFA

True

Explanation of Solution

Given:

Function: $g(x)=-\frac{1}{2}\left|x\right|+4$

Concept used:

The general form of an absolute value function is $f(x)=a\left|x-h\right|+k$ .

The graph opens up if $a>0$ and opens down if $a<0$ . If $0<a<1$ , the graph is compressed, and if $a>1$ , it is stretched.

The graph of $f(x)=\left|x\right|$ is translated $h$ units horizontally and $k$ units vertically to get the graph of $f(x)=a\left|x-h\right|+k$ .

The graph of $f(x)=\left|x\right|$ is compressed by $-\frac{1}{2}$ , translated $4$ units vertically up to get the graph of $g(x)=-\frac{1}{2}\left|x\right|+4$ .

Conclusion:

Therefore, the graph of $f(x)=\left|x\right|$ is translated $4$ units vertically up to get the graph of $g(x)=-\frac{1}{2}\left|x\right|+4$ .

Therefore the given statement is true.