Math 1111

1. Math 1111 Final Exam Review

2. 1. Identify the type of function. f(x) = 5 Constant

3. 2. Identify the type of function. Quotient of two polynomials. Rational Function

4. 3. Identify the type of function. Highest Exponent 3 Polynomial Function

5. 4. Identify the type of function. Highest Exponent 2 Quadratic Function

6. 5. Identify the type of function. Variable in the Exponent Exponential Function

7. 6. Identify the type of function. Linear

8. 7. $1500 is invested at a rate of 6¼% compounded continuously. What is the balance at the end of two years? • A = balance at the end of investment period • P = Principal (Money invested) • r = rate = 0.0625 • t = time in years = 2 Continuous Compounding $1699.73

9. 8. Evaluate the expression. • Round your answer to 3 decimal places. ≈ 26.565

10. 10. Evaluate the expression. • When t = 15. • Round your answer to 2 decimal places. ≈ 95.95

11. 11. What transformations are used to create the “child”? Parent Child 1. Reflection of x-axis. 2. Vertical shift of 5 units.

12. 12. What transformations are used to create the “child”? Parent Child 1. Reflection of y-axis. 2. Vertical shift up of 1 unit. 3. Horizontal shift right of 4 units.

13. 13. $2100 is invested at a rate of 7% compounded monthly. What is the balance at the end of 10 years? • A = balance at the end of investment period • P = Principal (Money invested) • r = rate = 0.07 • n = number of periods per year = 12 • t = number of years n Compoundings per year $4220.29

14. 15a. Given f(x) = x3 – 2x2 – 21x – 18answer the following questions. • What is the degree of the polynomial? 3 • According to the Fundamental Theorem of Algebra, how many zeros will this polynomial have? 3 • Use Descartes’ Rule of Signs to determine the number of possible positive real zeros. 1 possible positive zero, one sign variation • Use Descartes’ Rule of Signs to determine the number of possible negative real zeros. f(-x) = (-x)3 – 2(-x)2 -21(-x) – 18 = -x3 – 2x2 + 21x - 18 2 or 0 possible negative zeros, two sign variations

15. 15b. Given f(x) = x3 – 2x2 – 21x – 18answer the following questions. • Use the Rational Root Test to list All possible rational zeros of the polynomial. • Where p is a factor of -18 and q is a factor of 1

16. Given all of these candidates, how can you tell which is the actual root? f(x) = x3 – 2x2 – 21x – 18 Synthetic Division Graphing calculator

17. 15c. Given f(x) = x3 – 2x2 – 21x – 18answer the following questions. • Use the synthetic division to find all zeros of the polynomial. -1 18 -1 3 1 -3 -18 0 Each zero has a multiplicity of one.

18. 16a. Given f(x) = x4 + x3 – 11x2 + x – 12answer the following questions. • What is the degree of the polynomial? 4 • According to the Fundamental Theorem of Algebra, how many zeros will this polynomial have? 4 • Use Descartes’ Rule of Signs to determine the number of possible positive real zeros. 3 or 1 possible positive zero, three sign variations • Use Descartes’ Rule of Signs to determine the number of possible negative real zeros. f(-x) = (-x)4 + (-x)3 – 11(-x)2 + (-x) – 12 = x4 – x3 – 11x2 – x – 12 1 possible negative zero, one sign variation

19. 16b. Given f(x) = x4 + x3 – 11x2 + x – 12answer the following questions. • Use the Rational Root Test to list All possible rational zeros of the polynomial. • Where p is a factor of -12 and q is a factor of 1

20. 16c. Given f(x) = x4 + x3 – 11x2 + x – 12answer the following questions. 3 3 3 12 12 1 4 1 4 0 Factor by grouping Extract the root. Each zero has a multiplicity of one.

21. 17a. Find and plot the y-intercept. Write as an ordered pair. Set x = 0 y-intercept

22. 17b. Find and plot the zeros. Write as an ordered pair. Set f(x) = y = 0

23. 17. Vertical Asymptote Set the denominator = 0 x = 3

24. 17. Horizontal Asymptote y = 1 Since the degree of the two polynomials is the same find the ratio of the leading coefficient of the numerator divided by the leading coefficient of the denominator. x = 3

25. 17. Find function values to help you graph. y = 1 x = 3

26. 17. Find function values to help you graph. y = 1 x = 3

27. 18a. Find and plot the y-intercept. Write as an ordered pair. Set x = 0 y-intercept

28. 18b. Find and plot the zeros. Write as an ordered pairs. Set f(x) = y = 0

29. 18. Vertical Asymptote Set the denominator = 0 x = 3

30. 18. Horizontal Asymptote Since the degree of the numerator is greater than the degree of the denominator there is no horizontal asymptote. Slant Asymptote y = x + 4

31. 18. Find function values to help you graph. y =x + 4 x = 3

32. 19a. Find and plot the y-intercept. Write as an ordered pair. Set x = 0 y-intercept

33. 19b. Find and plot the zeros. Write as an ordered pairs. Set f(x) = y = 0 The Only Zero

34. 19. Vertical Asymptote Set the denominator = 0 x = -4 x = 1

35. 19. Horizontal Asymptote Since the degree of the numerator is less than the degree of the denominator the horizontal asymptote is y = 0.

36. 19. Find function values to help you graph. y = 0 x = -4 x = 1

37. 20. Find the Domain:

38. 21. Find the Domain:

39. 22. Match the function with the graph: • f(x) = 4x – 5 • f(x) = 4x + 5 • f(x) = 4-x + 5 • f(x) = 4-x – 5

40. 23. Match the function with the graph: • f(x) = 3x-1 • f(x) = 3x – 1 • f(x) = 31- x • f(x) = 3-x – 1

41. 24. Match the function with the graph: • f(x) = 5x+1 – 2 • f(x) = 5x+2 – 1 • f(x) = 5x-1+ 2 • f(x) = 5x-2 + 1