Chapter 3.5, Problem 4BGP

To determine

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

Answer to Problem 4BGP

The function $f\left(x\right)$ is first undergoes horizontal shift of 1 unit towards right and then reflected vertically about X-axis and then undergoes Vertical Stretch.

Explanation of Solution

Given information:

  $f\left(x\right)=x$

  $h\left(x\right)=-2.5\left(x-1\right)$

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.

Calculation:

From the given information,

  $f\left(x\right)=x$

  $h\left(x\right)=-2.5\left(x-1\right)$

  $\Rightarrow h\left(x\right)=-2.5\times f\left(x-1\right)$

Therefore, from above concepts

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.

Here $h=-1$

As $h<0$ it moves 1 unit towards right.

  $\Rightarrow h\left(x\right)=2.5\times \left(-f\left(x-1\right)\right)$

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

Therefore, it undergoes Vertical reflection.

  $\Rightarrow h\left(x\right)=2.5\times \left(-f\left(x-1\right)\right)$

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.

Here $a=2.5$ which is greater than 1

So, it undergoes Vertical stretch

Therefore, the function $f\left(x\right)$ is first undergoes horizontal shift of 1 unit towards right and then reflected vertically about X-axis and then undergoes Vertical Stretch.