Chapter 3.5, Problem 50PPS

a.

To determine

To find: Transformation of given graph of $f\left(x\right)=x$

Answer to Problem 50PPS

Graph 4

Explanation of Solution

Given information:

  $f\left(x\right)=x$

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

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.

From the given information,

  $g\left(x\right)=\frac{1}{2}\left(-x\right)$

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

Since the graph of $\left(-f\left(x\right)\right)$ is reflection of graph of $f\left(x\right)$ about x-axis.

And also $g\left(x\right)=\frac{1}{2}\left(-f\left(x\right)\right)$ comparing with above concept of $g\left(x\right)=a\cdot f\left(x\right)$

We get $a=\frac{1}{2}$ which is $0<\frac{1}{2}<1$

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

So, the graph of $\left(-f\left(x\right)\right)$ is compressed vertically.

Therefore, the graph of $g\left(x\right)=-\frac{1}{2}x$ is same as $f\left(x\right)=x$ which is reflected about x-axis and compressed vertically.

Therefore, the Graph 4 matches with the given graph.

b.

To determine

To find:Transformation of given graph of $f\left(x\right)=x$

Answer to Problem 50PPS

Graph 2

Explanation of Solution

Given information:

  $f\left(x\right)=x$

  $p\left(x\right)=2f\left(x\right)$

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.

From the given information,

  $p\left(x\right)=2x$

  $\Rightarrow p\left(x\right)=2f\left(x\right)$

Since the $p\left(x\right)=2f\left(x\right)$ comparing with above concept of $g\left(x\right)=a\cdot f\left(x\right)$

We get $a=2$ which is $2>1$

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

So, the graph of $f\left(x\right)$ is stretched vertically.

Therefore, the graph of $p\left(x\right)=2x$ is same as $f\left(x\right)=x$ which is stretched vertically.

Therefore, the Graph 2 matches with the given graph.

c.

To determine

To find:Transformation of given graph of $f\left(x\right)=x$

Answer to Problem 50PPS

Graph 1

Explanation of Solution

Given information:

  $f\left(x\right)=x$

  $r\left(x\right)=f\left(x\right)+2$

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.

From the given information,

  $r\left(x\right)=f\left(x\right)+2$

Since the $r\left(x\right)=f\left(x\right)+2$ comparing with above concept of $g\left(x\right)=f\left(x\right)+k$

We get $k=2$ which is $2>0$ .

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.

So, the graph of $f\left(x\right)$ is shifted (translated) vertically upwards of $2$ units.

Therefore, the graph of $r\left(x\right)=f\left(x\right)+2$ is same as $f\left(x\right)=x$ which is shifted (translated) vertically upwards of $2$ units.

Therefore, the Graph 1 matches with the given graph.

d.

To determine

To find: Transformation of given graph of $f\left(x\right)=x$

Answer to Problem 50PPS

Graph 3

Explanation of Solution

Given information:

  $f\left(x\right)=x$

  $j\left(x\right)=-f\left(x\right)$

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.

From the given information,

  $j\left(x\right)=-f\left(x\right)$

Since the $j\left(x\right)=-f\left(x\right)$ comparing with above concept of $g\left(x\right)=-f\left(x\right)$

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

So, the graph of $f\left(x\right)$ is reflected about x-axis.

Therefore, the graph of $j\left(x\right)=-f\left(x\right)$ is same as $f\left(x\right)=x$ which is reflected about x-axis.

Therefore, the Graph 3 matches with the given graph.