Chapter 3.5, Problem 59PFA

To determine

To find: Transformations of function $f\left(x\right)=x$ which makes less steep than it.

Answer to Problem 59PFA

The translations of function $f\left(x\right)=x$ which makes it less steep are:

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

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

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

Explanation of Solution

Given information:

  $f\left(x\right)=x$

  $\begin{array}{l}g\left(x\right)=f\left(x+2\right)\\ h\left(x\right)=f\left(x-2\right)\\ j\left(x\right)=f\left(\frac{1}{2}x\right)\\ k\left(x\right)=f\left(2x\right)\\ m\left(x\right)=f\left(-\frac{1}{2}x\right)\\ n\left(x\right)=\frac{1}{2}f\left(x\right)\end{array}$

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 $a>0$ 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,

  $f\left(x\right)=x$

  $g\left(x\right)=f\left(x+2\right)$ is translation of function $f\left(x\right)$ horizontally shifted towards left by 2 units.

  $\Rightarrow g\left(x\right)=x+2$

The slope of function $g\left(x\right)$ is same as of function $f\left(x\right)$ .

So, the translation does not make the function less steep.

  $h\left(x\right)=f\left(x-2\right)$ is translation of function $f\left(x\right)$ vertically shifted towards right by 2 units.

  $\Rightarrow h\left(x\right)=x-2$

The slope of function $h\left(x\right)$ is same as of function $f\left(x\right)$ .

So, the translation does not make the function less steep.

  $j\left(x\right)=f\left(\frac{1}{2}x\right)$ is translation of function $f\left(x\right)$ which is horizontally compressed.

  $\Rightarrow j\left(x\right)=\frac{1}{2}x$

The slope of function $j\left(x\right)$ is $\frac{1}{2}$ which is less than that of function $f\left(x\right)$ .

So, the translation makes the function less steep.

  $k\left(x\right)=f\left(2x\right)$ is translation of function $f\left(x\right)$ which is horizontally stretched.

  $\Rightarrow k\left(x\right)=2x$

The slope of the function $k\left(x\right)$ is 2 which is greater than that of function $f\left(x\right)$ .

So, the translation does not make the function less steep. (instead translation makes steeper)

  $m\left(x\right)=f\left(-\frac{1}{2}x\right)$ is translation of function $f\left(x\right)$ which is reflected first about Y-axis and then horizontally compressed.

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

The slope of the function $m\left(x\right)$ is $-\frac{1}{2}$ and its absolute value is $\frac{1}{2}$ which is less than that of slope of function $f\left(x\right)$ .

So, the translation makes the function less steep.

  $n\left(x\right)=\frac{1}{2}f\left(x\right)$ is translation of function $f\left(x\right)$ which is Vertically compressed.

  $\Rightarrow n\left(x\right)=\frac{1}{2}x$

The slope of the function $n\left(x\right)$ is $\frac{1}{2}$ which is less than the slope of function $f\left(x\right)$ .

So, the translation makes the function less steep.