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

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

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. n Compoundings per year
13. $2100 is invested at a rate of 7% compounded monthly. What is the balance at the end of 10 years?
n Compoundings per year
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 $

14. 15a. Given f(x) = x3 – 2x2 – 21x – 18 answer 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?
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 – 18 answer 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 – 18 answer the following questions.
Use the synthetic division to find all zeros of the polynomial. 1 -2
-21
-18 -1 18 -1 3 1 -3
-18
Each zero has a multiplicity of one.

18. 16a. Given f(x) = x4 + x3 – 11x2 + x – 12 answer 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?
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 – 12 answer 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 – 12 answer the following questions.
-11
-12 3 3 3 12 12 1 4 1 4
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.
x = 3

26. 17. Find function values to help you graph.
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.
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