Chapter 3.2, Problem 41E

(a)

To determine

To sketch:The graph of the function and calculated the area formed by the triangle and positive xaixs.

Answer to Problem 41E

The graph of the function is shown in figure (1) and the area formed by the triangle and positive x axis is $25\text{sq}\text{.unit}$ .

Explanation of Solution

The given function is,

  $f\left(x\right)=5-\left|x-6\right|$

The graph of the above function is shown below.

  

Figure (1)

The area calculated as,

  $\begin{array}{c}Area=\frac{1}{2}bh\\ =\frac{1}{2}\left(11-1\right)\left(5\right)\\ =25\text{sq}\text{.}\text{unit}\end{array}$

Therefore, the graph of the function is shown in figure (1) and the area formed by the triangle and positive x axis is $25\text{sq}\text{.unit}$ .

(b)

To determine

To sketch: The graph of the function and calculated the area formed by the triangle and positive x axis and compare with the graph of part (a) then make a conjecture about the area of the triangle formed.

Answer to Problem 41E

The graph of the function is shown in figure (2) and the area formed by the triangle and positive x axis is $50\text{sq}\text{.unit}$ which is twice the area of triangle in part (a) and the conjecture of area formed by $y=cf\left(x\right)$ is $Area=25c\text{sq}\text{unit}$

Explanation of Solution

The given function is,

  $f\left(x\right)=5-\left|x-6\right|$

And,

  $\begin{array}{c}y=2f\left(x\right)\\ =2\left(5-\left|x-6\right|\right)\\ =10-2\left|x-6\right|\end{array}$

The graph of the above function is shown below.

  

Figure (2)

The area calculated as,

  $\begin{array}{c}Area=\frac{1}{2}bh\\ =\frac{1}{2}\left(10\right)\left(10\right)\\ =50\text{sq}\text{.}\text{unit}\end{array}$

The area formed is double the area of triangle formed and in part (a).

The conjecture of area formed by $y=cf\left(x\right)$ is,

  $Area=25c\text{sq}\text{unit}$

Therefore, the graph of the function is shown in figure (2) and the area formed by the triangle and positive x axis is $50\text{sq}\text{.unit}$ which is twice the area of triangle in part (a) and the conjecture of area formed by $y=cf\left(x\right)$ is $Area=25c\text{sq}\text{unit}$

(c)

To determine

To sketch: The graph of the function and calculated the area formed by the triangle and positive x axis and compare with the graph of part (a) then make a conjecture about the area of the triangle formed.

Answer to Problem 41E

The graph of the function is shown in figure (3) and the area formed by the triangle and positive x axis is $25\text{sq}\text{.unit}$ which is same as the area of triangle in part (a) and the conjecture of area formed by $y=f\left(x\pm c\right)$ is $Area=25\text{sq}\text{unit}$

Explanation of Solution

The given function is,

  $f\left(x\right)=5-\left|x-6\right|$

And,

  $\begin{array}{c}y=f\left(x-3\right)\\ =5-\left|x-3-6\right|\\ =5-\left|x-9\right|\end{array}$

The graph of the above function is shown below.

  

Figure (3)

The area calculated as,

  $\begin{array}{c}Area=\frac{1}{2}bh\\ =\frac{1}{2}\left(10\right)\left(5\right)\\ =25\text{sq}\text{.}\text{unit}\end{array}$

The area formed is double the area of triangle formed and in part (a).

The conjecture of area formed by $y=f\left(x\pm c\right)$ is,

  $Area=25\text{sq}\text{unit}$

Therefore, the graph of the function is shown in figure (3) and the area formed by the triangle and positive x axis is $25\text{sq}\text{.unit}$ which is same as the area of triangle in part (a) and the conjecture of area formed by $y=f\left(x\pm c\right)$ is $Area=25\text{sq}\text{unit}$