1. Explain why the values of an increasing exponential function will eventually overtake the values of an increasing linear function.

2. Given a formula for an exponential function, is it possible to determine whether the function grows or decays exponentially just by looking at the formula? Explain.

For the following exercises, identify whether the statement represents an exponential function. Explain.

3. A population of bacteria decreases by a factor of [latex]\frac{1}{8}[/latex] every 24 hours.

4. The value of a coin collection has increased by 3.25% annually over the last 20 years.

5. For each training session, a personal trainer charges his clients $5 less than the previous training session.

6. The height of a projectile at time t is represented by the function [latex]h\left(t\right)=-4.9{t}^{2}+18t+40[/latex].

For the following exercises, consider this scenario: For each year t, the population of a forest of trees is represented by the function [latex]A\left(t\right)=115{\left(1.025\right)}^{t}[/latex]. In a neighboring forest, the population of the same type of tree is represented by the function [latex]B\left(t\right)=82{\left(1.029\right)}^{t}[/latex]. (Round answers to the nearest whole number.)

7. Which forest’s population is growing at a faster rate?

8. Which forest had a greater number of trees initially? By how many?

9. Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after 20 years? By how many?

10. Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after 100 years? By how many?

11. 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?

For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain.

12. [latex]y=300{\left(1-t\right)}^{5}[/latex]

13. [latex]y=220{\left(1.06\right)}^{x}[/latex]

14. [latex]y=16.5{\left(1.025\right)}^{\frac{1}{x}}[/latex]

15. [latex]y=11,701{\left(0.97\right)}^{t}[/latex]

For the following exercises, find the formula for an exponential function that passes through the two points given.

16. [latex]\left(0,6\right)[/latex] and [latex]\left(3,750\right)[/latex]

17. [latex]\left(0,2000\right)[/latex] and [latex]\left(2,20\right)[/latex]

18. [latex]\left(3,1\right)[/latex] and [latex]\left(5,4\right)[/latex]

For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain.

19. [latex]y=3742{\left(e\right)}^{0.75t}[/latex]

20. [latex]y=150{\left(e\right)}^{\frac{3.25}{t}}[/latex]

21. [latex]y=2.25{\left(e\right)}^{-2t}[/latex]

22. Suppose an investment account is opened with an initial deposit of $12,000 earning 7.2% interest compounded continuously. How much will the account be worth after 30 years?

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary.

23. [latex]f\left(x\right)=-{4}^{2x+3}[/latex], for [latex]f\left(-1\right)[/latex]

24. [latex]f\left(x\right)={e}^{x}[/latex], for [latex]f\left(3\right)[/latex]

25. [latex]f\left(x\right)=-2{e}^{x - 1}[/latex], for [latex]f\left(-1\right)[/latex]

26. [latex]f\left(x\right)=2.7{\left(4\right)}^{-x+1}+1.5[/latex], for [latex]f\left(-2\right)[/latex]

27. [latex]f\left(x\right)=1.2{e}^{2x}-0.3[/latex], for [latex]f\left(3\right)[/latex]

28. [latex]f\left(x\right)=-\frac{3}{2}{\left(3\right)}^{-x}+\frac{3}{2}[/latex], for [latex]f\left(2\right)[/latex]

29. The fox population in a certain region has an annual growth rate of 9% per year. In the year 2012, there were 23,900 fox counted in the area. What is the fox population predicted to be in the year 2020?

30. A scientist begins with 100 milligrams of a radioactive substance that decays exponentially. After 35 hours, 50mg of the substance remains. How many milligrams will remain after 54 hours?

31. In the year 1985, a house was valued at $110,000. By the year 2005, the value had appreciated to $145,000. What was the annual growth rate between 1985 and 2005? Assume that the value continued to grow by the same percentage. What was the value of the house in the year 2010?

32. A car was valued at $38,000 in the year 2007. By 2013, the value had depreciated to $11,000 If the car’s value continues to drop by the same percentage, what will it be worth by 2017?

For the following exercises, graph each set of functions on the same axes.

33. [latex]f\left(x\right)=3{\left(\frac{1}{4}\right)}^{x}[/latex], [latex]g\left(x\right)=3{\left(2\right)}^{x}[/latex], and [latex]h\left(x\right)=3{\left(4\right)}^{x}[/latex]

34. [latex]f\left(x\right)=\frac{1}{4}{\left(3\right)}^{x}[/latex], [latex]g\left(x\right)=2{\left(3\right)}^{x}[/latex], and [latex]h\left(x\right)=4{\left(3\right)}^{x}[/latex]

For the following exercises, match each function with one of the graphs pictured below.

35. [latex]f\left(x\right)=2{\left(0.69\right)}^{x}[/latex]

36. [latex]f\left(x\right)=2{\left(1.28\right)}^{x}[/latex]

37. [latex]f\left(x\right)=2{\left(0.81\right)}^{x}[/latex]

38. [latex]f\left(x\right)=4{\left(1.28\right)}^{x}[/latex]

39. [latex]f\left(x\right)=2{\left(1.59\right)}^{x}[/latex]

40. [latex]f\left(x\right)=4{\left(0.69\right)}^{x}[/latex]

For the following exercises, use the graphs shown below. All have the form [latex]f\left(x\right)=a{b}^{x}[/latex].

41. Which graph has the largest value for b?

42. Which graph has the smallest value for b?

43. Which graph has the largest value for a?

44. Which graph has the smallest value for a?

For the following exercises, graph the transformation of [latex]f\left(x\right)={2}^{x}[/latex]. Give the horizontal asymptote, the domain, and the range.

45. [latex]h\left(x\right)={2}^{x}+3[/latex]

46. [latex]f\left(x\right)={2}^{x - 2}[/latex]

For the following exercises, evaluate the exponential functions for the indicated value of x.

47. [latex]g\left(x\right)=\frac{1}{3}{\left(7\right)}^{x - 2}[/latex] for [latex]g\left(6\right)[/latex].

48. [latex]f\left(x\right)=4{\left(2\right)}^{x - 1}-2[/latex] for [latex]f\left(5\right)[/latex].

49. [latex]h\left(x\right)=-\frac{1}{2}{\left(\frac{1}{2}\right)}^{x}+6[/latex] for [latex]h\left(-7\right)[/latex].