In Problems 69–72, the graph of a function $f$is illustrated. Use the graph of $f$as the first step toward graphing each of the following functions:

$$\begin{gathered} \left(a\right)F\left(x\right)=f\left(x\right)+3 \\ \left(b\right)G\left(x\right)=f(x+2) \\ \left(c\right)P\left(x\right)=-f\left(x\right) \\ \left(d\right)H\left(x\right)=f(x+1)-2 \\ \left(e\right)Q\left(x\right)=\frac{1}{2}f\left(x\right) \\ \left(f\right)g\left(x\right)=f(-x) \\ \left(g\right)h\left(x\right)=f\left(2x\right) \end{gathered}$$

The graph:

To graph the function $F\left(x\right)=f\left(x\right)+3$

The graph of $F\left(x\right)$is obtained by adding $3$units to $f\left(x\right)$, so that the graph is shifted vertically upward to $3$units.

The graph:

To graph the given function

Replace $x$by $x+2$, so that the graph shifted horizontally right to $2$unit.

The graph:

To graph the given function

The graph of the function $P\left(x\right)=-f\left(x\right)$is reflected about $x-$axis, so, the graph is:

The graph:

To graph the given function.

The graph of $H\left(x\right)$is shifted horizontally right to $1$unit since $x$is replaced by $(x+1)$.

And the graph also shifted downward to $2$units, since $2$is subtracted from the function.

The graph:

To graph the given function.

Since the graph is multiplied by the factor $\frac{1}{2}$, the graph stretched by the factor $\frac{1}{2}$.

The graph:

To graph the given function.

The graph of the function is the reflection of the given function about $y-$axis. So,

The graph:

To graph the given function.

The function $h\left(x\right)$is obtained by each $x-$coordinate multiplied by the factor of $2$.