Example LF.1.3 :$f(x) =2x$.  "Scalar Multiple": $2$.
The arrows in the mapping diagram all end at the a point, $2x$, twice as far from the position $0$ as the position of  $x$ at the tail of the arrow .
Motion (Rate) Interpretation: If $x$ is time and $f(x)$ is the position of an object on the target axis at time $x$, then at time $0$ the object is at position $0$. The position at time $x$ is is precisely $2x$ units greater (less) then $0$ when $x>0$ ($x<0$). The object is moving at a velocity of $\frac {\text {2 units of position}} {\text {unit time}} = 2 \frac {\text {units of position}} {\text {unit time}}$.
Magnification Interpretation: If $\Delta x = |x_1-x_0|$ is the distance  between the points on the domain axis corresponding to numbers $x_0$ and $x_1$, then the distance between the points on the target axis corresponding to numbers $f(x_0) = 2x_0$ and $f(x_1) = 2x_1$ is
$\Delta y = |f(x_1)-f(x_0)| = |2x_1-2x_0| =2 |x_1-x_0| = 2 \Delta x$.
Thus the interval between the points $x_1$ and $x_0$ is magnified by a factor of $2$.
Slope Interpretation:  This interpretation is attached to the graph of $f$ in the coordinate plane. The slope is the $\frac {\text {rise}} {\text {run}} =\frac {\Delta y} {\Delta x}= 2$.