Chapter 2.5, Problem 68E

To determine

To sketch the graph of the function

Explanation of Solution

Given information:

The function is $y=〚x〛$

Concept used:

The greatest integer function is defined by,

  $〚x〛$ is the greatest integer less than or equal to $x$ .

The horizontal shrinking and stretching of graph are defined by,

To graph, $y=f\left(cx\right)$ ;

If $c>1$ , shrink the graph of $y=f\left(x\right)$ horizontally by a factor of $\frac{1}{c}$ .

If $0<c<1$ , stretch the graph of $y=f\left(x\right)$ horizontally by a factor of $\frac{1}{c}$ .

Graph:

To sketch the graph of the function $f\left(x\right)=〚x〛$ , substitute different value of $x$ and find the corresponding value of $f\left(x\right)=〚x〛$ .

The table showing the values of $f$ for some value of $x$ is given below,

$x$	$y=〚x〛$
$⋮$	$⋮$
$-2\le x<-1$	$-2$
$-1\le x<0$	$-1$
$0\le x<1$	$0$
$1\le x<2$	$1$
$2\le x<3$	$2$
$⋮$	$⋮$

The function is constant between consecutive integers, so the graph between integers is a horizontal line segment as shown below:

  

To graph $y=〚\frac{1}{4}x〛$ , use horizontal shrinking and stretching of graph,

Since, $0<c=\frac{1}{4}<1$ , so, the graph of $y=〚\frac{1}{4}x〛$ is obtained by stretching the graph of $y=〚x〛$ horizontally by a factor of $\frac{1}{4}$ .

Hence,

The graph of $y=〚\frac{1}{4}x〛$ is obtained as:

  

Interpretation:

From the above graph, it can be observed that graph of $y=〚\frac{1}{4}x〛$ is obtained by stretching the graph of $y=〚x〛$ horizontally by a factor of $\frac{1}{4}$ .