Find a possible formula for the exponential function with the points $(0, 30)$ and $(6, 5)$

Question

My son has the following problem he needs help with:

Find a possible formula for the exponential function with the following two points: $(0, 30)$ and $(6, 5)$. Equation Form = f(x)=a*b^x

Can somebody tell me how to do it so I can teach him?

Thanks

Answer 1

Assume $f(x)=Ae^x+B$ is such a function. Then we know $f(0)=30$ and $f(6)=5$. However, $f(0)=Ae^0+B=A+B$ and $f(6)=Ae^6+B$. This implies the following system of equations holds: $$ \begin{cases} A+B=30 \\ Ae^6+B=5 \end{cases} $$ Subtracting the equations gives $(A+B)-(Ae^6+B)=30-5$ or $A-Ae^6=25$ meaning $A(1-e^6)=25$ or $A=\frac{25}{1-e^6}$. From the first equation, we deduce $B=30-\frac{25}{1-e^6}$. We conclude $f(x)=\frac{25}{1-e^6}e^x+30-\frac{25}{1-e^6}=\frac{25}{1-e^6}(e^x-1)+30$ is such a function.

Answer 2

It depends on what you mean by "exponential". I'll assume you mean $f(x)=Ae^{Bx}$ with parameters $A,B$. As a good rule of thumb, you need two equations to get two unknowns and here you have two equations.

We get $$30=f(0)=Ae^0=A\implies A=30$$

and $$5=f(6)=30e^{6B}\implies e^{6B}=\frac 16\implies B=-\frac {\ln 6}6$$