How do I find the exponential equation from just its graph?

Question

I am posting here for the first time, so sorry if there are any formatting issues.

So I'm trying to find the equations in the form $f(x)=be^{-ax}+c$ for $b$, $c$ and $a$ for the two separate graphs below. The only one I know how to find is $c$ which is the horizontal asymptote, so for both the graphs equations, it would be $f(x)=be^{-ax}+20$ but I have no idea how to find $b$ and $c$. If someone could give me some pointers, then I could check by graphing them.

The following are the two graphs: ![Graph 1, listed with horizontal asymptote, y intercept and random point][1]

![Graph 2, listed with horizontal asymptote, y intercept and random point][2]

Answer 1

You can form 3 equation and solve for a,b and c (Just solving for graph 1)

For x tends to $\infty$

$ 20= be^{-a*\infty}+c$

$ 20=c$

Now for $x=0$

$22.5=be^{-a*0}+20$

$2.5=b$

For $x=1$

$20.616=2.5e^{-a*1}+20$

$0.2464=e^{-a}$

$Ln(0.2464)=-a$

$-1.4=-a$

$a=1.4$

Now $f(x) = 2.5e^{-1.4x}+20$

Graph from desmos

Answer 2

Plug the given values into the equation. If you plug in the $x$ and $y$ values at $x = 0$, you would get this equation:

$$b e^0 + 20 = 22.5$$

$$\implies b + 20 = 22.5 \implies b = 2.5$$

Using that $b$ value and the $x$ and $y$ values of the second point should give you an equation that can be solved with logarithms (for the a value).