Get In-Depth Help with Exponential models Solutions and Real-World Applications

Look at this table:
$\begin{array}{|cc|}\hline x& y\\ 1& 2\\ 2& 4\\ 3& 8\\ 4& 16\\ 5& 32\\ \hline\end{array}$
Write a linear $(y=mx+b)$, quadratic $(y=ax2)$, or exponential $(y=a(b)x)$ function that models the data. $y=$

You open a bank account to save for college and deposit $400 in the account. Each year, the balance in your account will increase 5%. a. Write a function that models your annual balance. b. What will be the total amount in your account after 7 yr? Recall that the general form of an exponential function is ${\displaystyle f\left(x\right)=a{b}^x}$.
Given the situation described, what is a? What is b? Write the function that models your annual balance. Check to make sure the function you created provides the same values you put into your table for years 1, 2 and 3. a = _____ b = _____ f(x) = _____

For the following exercises, consider this scenario: For each year t, the population of a forest of trees is represented by the function $A(t)=115(1.025{)}^t$. In a neighboring forest, the population of the same type of tree is represented by the function ${\displaystyle B\left(t\right)=82{\left(1.029\right)}^t}$. (Round answers to the nearest whole number.) Discuss the above results from the previous four exercises. Assuming the population growth models continue to represent the growth of the forests, which forest will have the greater number of trees in the long run? Why? What are some factors that might influence the long-term validity of the exponential growth model?

The polynomial ${\displaystyle -16t^2+vt+h0}$ models the height, in feet, of a ball tt seconds after it is thrown. In this polynomial, v represents the initial vertical velocity of the ball and h0 represents the initial height of the ball.
Min throws a baseball from a height of 6 feet with an initial vertical velocity of 30 feet per second. Which polynomial models the height of the ball, in feet?