How to graph y = 4 x + 3 using a table?

Consider it like this: You plug in something for ${\displaystyle x}$, and you follow the expression ${\displaystyle 4x+3}$ by multiplying by ${\displaystyle 4}$ then adding ${\displaystyle 3}$, to get the corresponding output value ${\displaystyle y}$.
This equation is linear: ${\displaystyle y=mx+b}$,
where ${\displaystyle m}$ is the slope (change in y over change in x) and ${\displaystyle b}$ is the y-intercept (where it hits the ${\displaystyle y}$ axis).

Your equation just has ${\displaystyle m=4}$ and ${\displaystyle b=3}$.
So, let's make a quick table. You need at least two points to make a straight line, and three points to make a curve. Let's do five points though for practice.

${\displaystyle x=-2\to y=4(-2)+3=-5}$
${\displaystyle x=-1\to y=4(-1)+3=-1}$
${\displaystyle x=0\to y=4\left(0\right)+3=3}$
${\displaystyle x=1\to y=4\left(1\right)+3=7}$
${\displaystyle x=2\to y=4\left(2\right)+3=11}$

In a table it then looks like this:

${\displaystyle \left[\begin{array}{ccc}\underline{y}& \underline{\text{|}}& \underline{x}\\ -5& \mid & -2\\ -1& \mid & -1\\ 3& \mid & 0\\ 7& \mid & 1\\ 11& \mid & 2\end{array}\right]}$

And after plotting each point by looking at ${\displaystyle x}$ and ${\displaystyle y}$, and moving to the right ${\displaystyle x}$ units and up ${\displaystyle y}$ units (if negative, move the opposite direction), we get:
graph{4x + 3 [-2, 2, -5, 11]}
Every point we put on the table should be visible if you look at this graph.