Chapter 1.7, Problem 26E

(a)

To determine

To graph: The scatter plot of the data with $t=6$ corresponding to $2006$ and identify two sets of points in the scatter plot that are approximately linear.

Answer to Problem 26E

The two sets of points in the scatter plot that are linear are $\left\{\left(2006,1488\right),\left(2007,1591\right),\left(2008,1682\right)\right\}$ and $\left\{\left(2009,1740\right),\left(2010,1750\right),\left(2011,1763\right),\left(2012,1778\right)\right\}$ .

Explanation of Solution

Given information:

The table shows the numbers $T$ of stores owned by the Target Corporation from $2006$ through $2012$ .

Year	Number of stores, $T$
$2006$	$1488$
$2007$	$1591$
$2008$	$1682$
$2009$	$1740$
$2010$	$1750$
$2011$	$1763$
$2012$	$1778$

Graph:

To graph the points on scatter plot, follow the steps using graphing calculator.

First press the $\boxed{2\text{nd}}$ key and then $\boxed{\text{Y}=}$ and make sure that Plot $1$ is ON and choose the scatter plot icon in type.

Go to ${\text{Y}}_1$ and then press the key $\boxed{\text{STAT}}$ and then in edit enter the data in ${\text{L}}_1$ and ${\text{L}}_2$ .

Now, press the $\boxed{\text{ZOOM}}\boxed{9}$ key and $\boxed{\text{TRACE}}$ key to produce the graph in standard window as shown in Figure (1).

  

Figure (1)

The points $\left(2006,1488\right)$ , $\left(2007,1591\right)$ , $\left(2008,1682\right)$ are approximately linear and the points $\left(2009,1740\right)$ , $\left(2010,1750\right)$ , $\left(2011,1763\right)$ and $\left(2012,1778\right)$ are approximately linear.

Therefore, the two sets of points in the scatter plot that are linear are $\left\{\left(2006,1488\right),\left(2007,1591\right),\left(2008,1682\right)\right\}$ and $\left\{\left(2009,1740\right),\left(2010,1750\right),\left(2011,1763\right),\left(2012,1778\right)\right\}$ .

(b)

To determine

To find: The linear model for each set of points with the help of regression of graphing utility.

Answer to Problem 26E

The linear model for the first set of points is equal to $T=97t+908$ and for second set of points is equal to $T=12.7t+1624.4$ .

Explanation of Solution

Given information:

The table shows the numbers $T$ of stores owned by the Target Corporation from $2006$ through $2012$ .

Year	Number of stores, $T$
$2006$	$1488$
$2007$	$1591$
$2008$	$1682$
$2009$	$1740$
$2010$	$1750$
$2011$	$1763$
$2012$	$1778$

Calculation:

For the regression linear model of data follow the steps below:

First press the $\boxed{2\text{nd}}$ key and then $\boxed{\text{Y}=}$ and make sure that Plot $1$ is ON and choose the scatter plot icon in type.

Go to ${\text{Y}}_1$ and then press the key $\boxed{\text{STAT}}$ and then in edit enter the data for first set in ${\text{L}}_1$ and ${\text{L}}_2$ .

Now enter the keystrokes $\boxed{\text{STAT}}\boxed{\to }\boxed{4}$ and press the $\boxed{\text{ENTER}}$ key for the regression.

  

Figure(2)

Therefore, the linear model for first set of points is equal to $T=97t+908$ .

For the regression linear model of data follow the steps below:

First press the $\boxed{2\text{nd}}$ key and then $\boxed{\text{Y}=}$ and make sure that Plot $1$ is ON and choose the scatter plot icon in type.

Go to ${\text{Y}}_1$ and then press the key $\boxed{\text{STAT}}$ and then in edit enter the data for second set in ${\text{L}}_1$ and ${\text{L}}_2$ .

Now enter the keystrokes $\boxed{\text{STAT}}\boxed{\to }\boxed{4}$ and press the $\boxed{\text{ENTER}}$ key for the regression.

  

Figure(3)

Therefore, the linear model for second set of points is equal to $T=12.7t+1624.4$ .

(c)

To determine

To find: The piecewise defined model for the data and the graph of it.

Answer to Problem 26E

The piecewise defined model for the data is given by $T=\{\begin{array}{l}97t+908,6\le t\le 8\\ 12.7t+1624.4,9\le t\le 12\end{array}$ and the graph of the model is given in Figure(4).

Explanation of Solution

Given information:

The table shows the numbers $T$ of stores owned by the Target Corporation from $2006$ through $2012$ .

Year	Number of stores, $T$
$2006$	$1488$
$2007$	$1591$
$2008$	$1682$
$2009$	$1740$
$2010$	$1750$
$2011$	$1763$
$2012$	$1778$

Calculation:

As the linear model for the first set of point is equal to $T=97t+908$ and the linear model for the second set of points is equal to $T=12.7t+1624.4$ .

So, the piecewise defined model is $T=\{\begin{array}{l}97t+908,6\le t\le 8\\ 12.7t+1624.4,9\le t\le 12\end{array}$ .

To graph a function $T=\{\begin{array}{l}97t+908,6\le t\le 8\\ 12.7t+1624.4,9\le t\le 12\end{array}$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=\{\begin{array}{l}97t+908,6\le t\le 8\\ 12.7t+1624.4,9\le t\le 12\end{array}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{97}\boxed{\text{X}}\boxed{+}\boxed{908}\boxed{)}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{4}\boxed{6}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{\to }\boxed{1}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{6}\boxed{8}\boxed{)}$ . Now press Enter after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{(}\boxed{12.7}\boxed{\text{X}}\boxed{+}\boxed{1624.4}\boxed{)}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{4}\boxed{9}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{\to }\boxed{1}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{6}\boxed{12}\boxed{)}$ .

The display will show the equation,

  $\begin{array}{c}{\text{Y}}_{\text{1}}=\left(97\text{X}+908\right)/\left(\text{X}\ge 6\text{and X}\le \text{8}\right)\\ {\text{Y}}_2=\left(12.7\text{X}+1624.4\right)/\left(\text{X}\ge 9\text{and X}\le 12\right)\end{array}$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown in Figure (1).

  

Figure (4)

(d)

To determine

To find: The scenario that could be the cause of break in data.

Answer to Problem 26E

One of the reason can be the shutdown of the business after $2008$ and then again starting it after $1$ year.

Explanation of Solution

Given information:

The table shows the numbers $T$ of stores owned by the Target Corporation from $2006$ through $2012$ .

Year	Number of stores, $T$
$2006$	$1488$
$2007$	$1591$
$2008$	$1682$
$2009$	$1740$
$2010$	$1750$
$2011$	$1763$
$2012$	$1778$

Calculation:

As there is a break in the graph between two lines from $2008$ and $2009$ .

Therefore, one of the reason can be the shutdown of the business after $2008$ and then again starting it after $1$ year.