Chapter 1, Problem 11PS

(a)

To determine

The graph of the function $H\left(x\right)-2$

Answer to Problem 11PS

The graph is plotted with vertical shift 2 units downwards.

Explanation of Solution

Given information:

Let us consider a Heaviside function

  $H\left(x\right)=\{\begin{array}{cc}1,& x\ge 0\\ 0,& x<0\end{array}$

And its graph is

  

  $H\left(x\right)-2$

Calculation:

The graph of $H\left(x\right)-2$ will be a vertical shift 2 units downward of the graph of $H\left(x\right)$ .

The graph is plotted below.

  

Conclusion:

The graph is plotted with vertical shift 2 units downwards.

(b)

To determine

The graph of the function $H(x-2)$

Answer to Problem 11PS

The graph of $H\left(x-2\right)$ will be plotted with horizontal shift 2 units rightwards.

Explanation of Solution

Given information:

Let us consider a Heaviside function

  $H\left(x\right)=\{\begin{array}{cc}1,& x\ge 0\\ 0,& x<0\end{array}$

And its graph is

  

  $H\left(x-2\right)$

Calculation:

The graph of $H\left(x-2\right)$ will be a horizontal shift 2 right of the graph of $H\left(x\right)$

The graph is plotted below.

  

Conclusion:

The graph of $H\left(x-2\right)$ will be plotted with horizontal shift 2 units rightwards.

(c)

To determine

The graph of the function $-H\left(x\right)$

Answer to Problem 11PS

The graph of $-H\left(x\right)$ will be a reflection in x -axis of the graph of $H\left(x\right)$

Explanation of Solution

Given information:

Let us consider a Heaviside function

  $H\left(x\right)=\{\begin{array}{cc}1,& x\ge 0\\ 0,& x<0\end{array}$

And its graph is

  

  $-H\left(x\right)$

Calculation:

The graph of $-H\left(x\right)$ will be a reflection in x -axis of the graph of $H\left(x\right)$

The graph is plotted below.

  

Conclusion:

The graph of $-H\left(x\right)$ will be a reflection in x -axis of the graph of $H\left(x\right)$

(d)

To determine

The graph of the function $H\left(-x\right)$

Answer to Problem 11PS

The graph of $H\left(-x\right)$ will be a reflection in y -axis of the graph of $H\left(x\right)$

Explanation of Solution

Given information:

Let us consider a Heaviside function

  $H\left(x\right)=\{\begin{array}{cc}1,& x\ge 0\\ 0,& x<0\end{array}$

And its graph is

  

  $H\left(-x\right)$

Calculation:

The graph of $H\left(-x\right)$ will be a reflection in y -axis of the graph of $H\left(x\right)$

The graph is plotted below.

  

Conclusion:

The graph of $H\left(-x\right)$ will be a reflection in y -axis of the graph of $H\left(x\right)$

(e)

To determine

The graph of the function $\frac{1}{2}H\left(x\right)$

Answer to Problem 11PS

The graph of $\frac{1}{2}H\left(x\right)$ will be a vertical shrink by $\frac{1}{2}$ of the graph of $H\left(x\right)$

Explanation of Solution

Given information:

Let us consider a Heaviside function

  $H\left(x\right)=\{\begin{array}{cc}1,& x\ge 0\\ 0,& x<0\end{array}$

And its graph is

  

  $\frac{1}{2}H\left(x\right)$

Calculation:

The graph of $\frac{1}{2}H\left(x\right)$ will be a vertical shrink by $\frac{1}{2}$ of the graph of $H\left(x\right)$

The graph is plotted below.

  

Conclusion:

The graph of $\frac{1}{2}H\left(x\right)$ will be a vertical shrink by $\frac{1}{2}$ of the graph of $H\left(x\right)$

(f)

To determine

The graph of the function $-H\left(x-2\right)+2$

Answer to Problem 11PS

The graph is obtained based on horizontal shift, reflection, vertical shift of 2 units upward.

Explanation of Solution

Given information:

Let us consider a Heaviside function

  $H\left(x\right)=\{\begin{array}{cc}1,& x\ge 0\\ 0,& x<0\end{array}$

And its graph is

  

  $-H\left(x-2\right)+2$

Calculation:

The graph of $-H\left(x-2\right)+2$ can be obtained from $H\left(x\right)$ by the following steps.

1) Horizontal shift by 2 unit’s right $\left(H\left(x-2\right)\right)$

2) Reflection in x −axis $\left(-H\left(x-2\right)\right)$

3) Vertical shifts 2 units upward $\left(-H\left(x-2\right)+2\right)$

The graph is plotted below.

  

Conclusion:

The graph is obtained based on horizontal shift, reflection, vertical shift of 2 units upward.