based off laws of exponents why does 2^x+3 and 8(2^x) have a relationship between graphs ?

Answer 1

Comparing 2^x+3 and 8(2^x), although they do not look alike, they are both the same.

Consider 8(2^x).

8 = 2³ so 8(2^x) = (2³)(2^x), The rules of exponents say when multiplying two expressions with the same base, you add the exponents so (2³)(2^x) = 2^x+3

So y = 2^x+3 and y = 8(2^x) have the same graph because they are identical.

Answer 2

There is a useful concept called something like the "root equation".....

Example:

The basic equation of a parabola is y = x²

This starts at the point(0,0) and "opens up"

If you modify the equation with a multiplier and/or offsets, the function still has the same basic shape when plotted.

In your example, the basic equation is y = n^x This is called the "exponential form". With the exception of n =1, the basic shape is similar regardless of n or any offsets.

To see all this, use an online graphing utility such as this one:

https://www.desmos.com/calculator