Chapter 3.5, Problem 56HP

a.

To determine

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

Answer to Problem 56HP

The function $f\left(x\right)$ is vertically shifted 3 units upwards along Y-axis to get $g\left(x\right)$

Explanation of Solution

Given information:

  $f\left(x\right)=x$

  $g\left(x\right)=x+3$

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.

We know that,

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.

From the given information,

  $g\left(x\right)=x+3$

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

Since $3>0$ , the function $f\left(x\right)$ is shifted vertically upwards by 3 units.

Therefore, the function $f\left(x\right)$ is Vertically shifted 3 units upwards along Y-axis to get $g\left(x\right)$ .

b.

To determine

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

Answer to Problem 56HP

The function $f\left(x\right)$ is Vertically reflected i.e. about X-axis to get $h\left(x\right)$ .

Explanation of Solution

Given information:

  $f\left(x\right)=x$

  $h\left(x\right)=-x$

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.

We know that,

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

From the given information,

  $h\left(x\right)=-x$

  $\Rightarrow h\left(x\right)=-f\left(x\right)$

Therefore, the function $f\left(x\right)$ is reflected about X-axis to obtain $h\left(x\right)$ .

c.

To determine

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

Answer to Problem 56HP

The function $k\left(x\right)$ is obtained by first shifting the function $f\left(x\right)$ horizontally towards right by 1 unit and then shifting vertically upwards by 5 units.

Explanation of Solution

Given information:

  $f\left(x\right)=x$

  $k\left(x\right)=\left(x-1\right)+5$

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.

We know that,

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.

From the given information,

  $k\left(x\right)=\left(x-1\right)+5$

  $\Rightarrow k\left(x\right)=f\left(x-1\right)+5$

Since $-1<0$ , the function $f\left(x\right)$ is shifted horizontally towards right by 1 unit to obtain $f\left(x-1\right)$

We know that,

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 $5>0$ , the obtained function $f\left(x-1\right)$ is then shifted vertically upwards by 5 units to obtain $k\left(x\right)$ .

Therefore, the function $k\left(x\right)$ is obtained by first shifting the function $f\left(x\right)$ horizontally towards right by 1 unit and then shifting vertically upwards by 5 units.