The easiest way to graph exponential functions is to simply find a few (x,y)(x,y) pairs that are easy to plot.
Let's start with y=3^xy=3x.
When x=0x=0 (i.e. when we're on the yy-axis), y=3^0=1y=30=1. So the point (0,1)(0,1) is on our graph. From here, every time xx goes up by 11, yy will triple. That is, for every one step right, our step up is 3 times larger than it was before.
x=1" "=>" "y=3*1=color(blue)3x=1 ⇒ y=3⋅1=3
x=2" "=>" "y=3*color(blue)3=color(green)9x=2 ⇒ y=3⋅3=9
x=3" "=>" "y=3*color(green)9=27x=3 ⇒ y=3⋅9=27
and so on.
graph{3^x [-17.15, 8.16, -1.75, 10.9]}
For the graph of y=3^(x+1)y=3x+1, we can actually recognize this as a translation of the first graph. The xx-value is being increased by 11; what this means is, at every xx-value, y=3^(x+1)y=3x+1 looks to the right of y=3^xy=3x by one unit, finds the yy-value there, and makes that its own yy-value at the same xx. This results in the original graph of y=3^xy=3x getting shifted to the left by one unit.
Think of it this way: when you look to your left, what was in your field of vision "moves right". Similarly, when an xx-input decreases by hh (a.k.a. (x-h)(x−h)), the parent graph shifts right by hh. That's why all these math formulas like y=asin[b(x-h)]+ky=asin[b(x−h)]+k and y=a(x-h)^2+ky=a(x−h)2+k have "-h−h" instead of "+h+h": because shifting a graph to the right requires a decrease in xx input, not an increase.
In short: y=3^(x+1)y=3x+1 will look identical to the graph of y=3^xy=3x, just shifted one unit to the left.
graph{3^(x+1) [-17.15, 8.16, -1.75, 10.9]}
