Question

if you answer this question i need an explanation so i can do the other questions on my own.write an exponential function whose graph passes through the given points.(0, 3) and (3, 375)

Answer 1

hello

to solve this problem, we have to take individual point and test it with standard equation of an exponential

$y=ae^{\alpha x}$

now we work with the first point which is (0, 3)

put y = 3 and x = 0 in the equation

$\begin{gathered} y=ae^{\alpha x} \\ 3=ae^{\alpha(0)} \\ 3=a\cdot1 \\ a=3 \end{gathered}$

now we know a = 3

we can test the second point which is (3, 375)

x = 3, y = 375 and a = 3

$\begin{gathered} y=ae^{\alpha x} \\ 375=3ae^{\alpha(3)} \\ \text{divide both sides by 3} \\ 125=e^{3\alpha} \\ \text{now we take the natural log of both sides} \\ In(125)=3\alpha \\ 3In5=3\alpha \\ \text{divide both sides by 3} \\ \frac{3In5}{3}=\frac{3\alpha}{3} \\ \alpha=In5 \end{gathered}$

now we can rewrite the equation of exponential function

$y=3ae^{In(5)(x)}$

we can go ahead to plot the graph now