Chapter 3, Problem 14PFA

a.

To determine

To find:Vertex of the graph.

Answer to Problem 14PFA

Vertex = $\left(2,1\right)$

Explanation of Solution

Given information:

  $g\left(x\right)=3\left|x-2\right|+1$

  $g\left(x\right)=3\left|x-2\right|+1$

Graph of given equation is

  

From the graph the slope changes at the point $\left(2,1\right)$ .

Hence the vertex of the graph is $\left(2,1\right)$ .

In general, the graph of absolute value function $f\left(x\right)=a\left|x-h\right|+k$ is a ‘V’ with vertex $\left(h,k\right)$ .

b.

To determine

To find:Translation of given equation relating to graph of $f\left(x\right)=\left|x\right|$ .

Answer to Problem 14PFA

Function $f\left(x\right)$ is shifted horizontally towards right by 2 units, stretched vertically and shifted vertically upward by 1 unit to obtain $g\left(x\right)$ .

Explanation of Solution

Given information:

  $\begin{array}{l}f\left(x\right)=\left|x\right|\\ g\left(x\right)=3\left|x-2\right|+1\end{array}$

Concept:

A Translation is a movement of the graph either horizontally parallel to x-axis or vertically parallel to y-axis.

The graph of $y=f\left(x\right)$ where $f\left(x\right)=x$ is same as the graph of $y=x$ . Writing the equations as functions in the form of $f\left(x\right)$ is useful when applying translations and reflections to graphs.

If $g\left(x\right)=f\left(x\right)+k$ it is known as vertical shift.

If $k>0$ it moves upwards.

If $k<0$ it moves downwards.

If $g\left(x\right)=f\left(x+h\right)$ it is known as horizontal shift.

If $h<0$ moves it right.

If $h>0$ moves it left.

If $g\left(x\right)=-f\left(x\right)$ it is known as Vertical reflection it means reflects about X-axis

If $g\left(x\right)=f\left(-x\right)$ it is known as Horizontal reflection it means reflects about Y-axis

If $g\left(x\right)=af\left(x\right)$

If $a>1$ it is known as Vertical stretch.

If $0<a<1$ it is known as Vertical compression.

If $g\left(x\right)=f\left(bx\right)$

If $b>1$ it is known as Horizontal compression.

If $0<b<1$ it is known as Horizontal Stretch.

  $g\left(x\right)=3\left|x-2\right|+1$

From given information

  $\Rightarrow g\left(x\right)=3f\left(x-2\right)+1$

From above concept

If $g\left(x\right)=f\left(x+h\right)$ it is known as horizontal shift.

If $h<0$ moves it right.

If $h>0$ moves it left.

Since $-2<0$ ,the function $f\left(x\right)$ is shifted horizontally towards right by 2 units to obtain $f\left(x-2\right)$ .

  $\Rightarrow g\left(x\right)=3f\left(x-2\right)+1$

From above concept

If $g\left(x\right)=af\left(x\right)$

If $a>1$ it is known as Vertical stretch.

If $0<a<1$ it is known as Vertical compression.

Since $3>1$ , the function $f\left(x-2\right)$ is stretched vertically to obtain $3f\left(x-2\right)$ .

  $\Rightarrow g\left(x\right)=3f\left(x-2\right)+1$

From above concept

If $g\left(x\right)=f\left(x\right)+k$ it is known as vertical shift.

If $k>0$ it moves upwards.

If $k<0$ it moves downwards.

Since $1>0$ , the function $3f\left(x-2\right)$ is shifted vertically upwards by 1 unit to obtain $g\left(x\right)$ .

c.

To determine

To find:Piecewise function of $g\left(x\right)$ .

Answer to Problem 14PFA

  $g\left(x\right)=-3x+7$ for $x<2$

  $g\left(x\right)=3x-5$ for $x>2$

Explanation of Solution

Given information:

  $g\left(x\right)=3\left|x-2\right|+1$

  $g\left(x\right)=3\left|x-2\right|+1$

For $x<2$ ,

  $\left|x-2\right|=-\left(x-2\right)$

Therefore

We get

  $\begin{array}{l}g\left(x\right)=3\left|x-2\right|+1\\ \Rightarrow g\left(x\right)=3\left(-\left(x-2\right)\right)+1\\ \Rightarrow g\left(x\right)=3\left(-x+2\right)+1\\ \Rightarrow g\left(x\right)=-3x+6+1\\ \Rightarrow g\left(x\right)=-3x+7\end{array}$

For $x>2$

  $\left|x-2\right|=x-2$

Therefore

We get

  $\begin{array}{l}g\left(x\right)=3\left|x-2\right|+1\\ \Rightarrow g\left(x\right)=3\left(x-2\right)+1\\ \Rightarrow g\left(x\right)=3x-6+1\\ \Rightarrow g\left(x\right)=3x-5\end{array}$

Therefore

  $g\left(x\right)=-3x+7$ for $x<2$

  $g\left(x\right)=3x-5$ for $x>2$