Answer
See image for the graphs.
Work Step by Step
$ H(x) = \left\{
\begin{array}{lr}
1 & : x \geq 0\\
0 & : x < 0
\end{array}
\right.$
(a) $ H(x)-2 = \left\{
\begin{array}{lr}
-1 & : x \geq 0\\
-2 & : x < 0
\end{array}
\right.$
This is found by subtracting 2 from each of the outputs of the Heaviside function. $$1-2=-1$$ $$0-2=-2$$
(b) $ H(x-2) = \left\{
\begin{array}{lr}
1 & : x \geq 2\\
0 & : x < 2
\end{array}
\right.$
This is found by substituting $(x-2)$ in for $x$ in the Heaviside function and then solving for $x$. $$x-2\geq0 \text{ thus } x\geq 2 $$ $$x-2<0 \text{ thus } x< 2$$
(c) $ -H(x) = \left\{
\begin{array}{r}
-1 & : x \geq 0\\
0 & : x < 0
\end{array}
\right.$
This is found by multiplying each of the outputs of the Heaviside function by -1. $$-1(1)=-1$$ $$-1(0)=0$$
(d) $ H(-x) = \left\{
\begin{array}{lr}
1 & : x \leq 0\\
0 & : x > 0
\end{array}
\right.$
This is found by substituting $-x$ in for $x$ in the Heaviside function and then solving for $x$. $$-x\geq0 \text{ thus } x\leq 0 $$ $$-x<0 \text{ thus } x>0$$
(e) $\frac{1}{2} H(x) = \left\{
\begin{array}{lr}
\frac{1}{2} & : x \geq 0\\
0 & : x < 0
\end{array}
\right.$
This is found by multiplying each of the outputs of the Heaviside function by $\frac{1}{2}$. $$\frac{1}{2}(1)=\frac{1}{2}$$ $$\frac{1}{2}(0)=0$$
(f) $ -H(x-2)+2 = \left\{
\begin{array}{lr}
1 & : x \geq 2\\
2 & : x < 2
\end{array}
\right.$
This one has two parts. First, substitute $(x-2)$ in for $x$ in the Heaviside function and solve for $x$. $$x-2\geq0 \text{ thus } x\geq 2 $$ $$x-2<0 \text{ thus } x< 2$$ Second, multiply each of the outputs of the Heaviside function by $-1$ then add 2. $$-1(1)+2=1$$ $$-1(0)+2=2$$
