math 55 intermediate algebra t....

Math 55 Intermediate Algebra T. Henson

The Final Exam will be given on Tuesday, December 17, 1:30 – 3:20 pm. NOTE that the final exam time is different from the lecture time. You will have the entire class time for the exam. It will cover the content specified below. Cell phones must be off during the exam. You may use a scientific calculator (graphing calculators not allowed). When you have finished taking the exam, you may leave. I will have the exams graded and course grades determined by Friday, Dec. 20. If you wish to know your final exam score and course grade, you can email me ([email protected]) and I will reply.

STUDY GUIDE and PRACTICE PROBLEMS A. Write in simplest form:

Rational expressions Expressions written with rational exponents Radical expressions (rationalize denominator as needed)

Practice Problems: Write each expression in simplest form. Rationalize denominators if needed. 1.

2

5

25

x

x

2. 2

2

3 10 8

3 2

x x

x x

3. 3/224z

4. 2/33

6

x

y

5. 3 632a b 6. 7 63 625m n

7. 4

10

9a

b

8. 4

3

16a

b

9. 2 3

3 5

B. Operations. Perform operations (add, subtract, multiply, divide) with the following types of

mathematical objects and express answers in simplest form: Rational expressions Expressions written with rational exponents Radical expressions (rationalize denominator as needed) Complex numbers Logarithmic expressions (add, subtract, use power rule)

Practice Problems: Perform the indicated operations and write in simplest form. Rationalize denominators if needed. 1.

2 2

2 12

36 36

x

x x

2. 2

2 1

1 1

x

x x

3. 5

3 2

x

x x

4. 2 2

2

5 6 5

1 25

x x x x

x x

5. 2 2

2 2

7 28

3 4 8 16

x x x x

x x x x

6. 12 1/31/3 1/4 3/4 3x y x y

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-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10 f(x)

x

y

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10 g(x)

x

y

7. 2 43 325 5m m 8. 5 2

5

27

54

x y

xy

9. 2 32

8

a a

b b

10. 16 7 12 11i i 11. 3 2i i 12. 3

2

i

i

Write as a single logarithm with coefficient of one.

13. 2log 3logx y 14. 1log 2log

3x y

C. Functions.

Evaluate functions, using appropriate function notation. Find the domain symbolically. Given the graph or table for a function, find the domain and

range. Use interval notation to write the domain and range. Perform operations with functions (add, subtract, multiply, divide) numerically or symbolically. Find the composition of two functions. Find the inverse of a function.

Practice Problems:

For the functions 2

1

1f x

x

and 1g x x ,

1. find 5f and 3g 2. find 4 8f g

3. find f x g x 4. find the domain of f (use interval notation)

5. find the domain of g (use interval notation) 6. find f g x

7. find 1g x

Use the graphs to determine the answers to the following questions. 8. Determine the domain and range of g. (use interval notation) 9. find 0fg 10.

find 2f

g

11. Find 2f g 12. Find 1 7f

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D. Solving Equations and Inequalities. Solve rational, absolute value, quadratic, higher-order polynomial, square root, cube root, exponential and logarithmic equations. Use factoring, the square root property and the quadratic formula to solve quadratic equations.

Practice Problems: Find all real or complex solutions for each equation. Express answers in exact, simplified form. Answers containing radicals should be simplified and rationalized as needed. Identify extraneous solutions. 1. 1 2

11 1

x

x x

2. 5 3 1

1 2x x

3. 5 3 14x

4. 2 1 5x x 5. 3 4 2x 6. 5 2 3x x

7. 6x x 8. 2 20 0x x 9. 23 5 20x

10. 2 8 5 0x x 11. 22 10 0x 12. 2/3 1/39 8 0x x

13. 2 12 32x 14. 10 500x 15. 53 27x

16. 3 7x 17. 3 1 28xe 18. 7 12xe

19. 2log 4 3x 20. log log 4 log 12x x 21. log 1 log 3 5x x

Solve compound and double inequalities. Express the solutions using interval notation. Graph the solution set on the number line. 20. 1 3 and 1 1x x 21. 2 7 5 and - 2 6x x

22. 5 1 3 or 1- 1x x 23. 9 5 4 5x

Solve systems of equations by using either the elimination/substitution method or by writing the augmented matrix of the system and using Gauss-Jordan elimination to reduce the matrix. Write the solution as an ordered triple. 24. Solve the system by using elimination and substitution. 2 5

2 7

2 7

x y

x y z

x y z

25. Write the system as an augmented matrix. Then use Gauss-Jordan elimination to solve the system.

2 1

2 5

2 10

x y z

x y z

y z

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E. Graphing functions and conic sections. Sketch the graphs of linear, absolute value, quadratic, rational, square root, exponential or logarithmic functions. Graphs should be appropriately scaled and labeled.

Functions. Practice Problems: Sketch the graph of each of the following.

State the domain and range using interval notation. For linear and quadratic functions, find the y intercept and find the x-intercept, if it exists. For square root and exponential functions, find the y-intercept.

1. 4 3f x x 2. 1f x x 3. 1

2f x

x

4. 2 4f x x 5. 2 2 3f x x x 6. 4f x x

7. /23xf x 8. 3logf x x

Conic Sections. Graph parabolas with horizontal axis of symmetry, circles, ellipses and hyperbolas. Practice Problems: Sketch the graph of each of the conic sections.

Graph the parabolas: For parabolas, state the coordinates for the vertex, the equation of the axis of symmetry and plot

at least two points in addition to the vertex. 9. 2 1x y 10. 2 2 3x y y

Graph the circles: For circles give the coordinates of the center and the radius. 12. 22 2 16x y 13. 2 2

3 2 9x y 14. 2 24 6 12x x y x

Graph the ellipses: For ellipses, state the coordinates of the vertices and endpoints of the minor axis. 15. 2 2

116 9

x y

16. 2 29 4 36x y

Graph the hyperbolas: For hyperbolas, state the coordinates of the vertices and include the asymptotes in your sketch. 17. 2 2

19 4

x y

18. 2 2

14 4

y x

F. Modeling and Applications.

1. Explain what a function is. Explain what it is that makes a function different from something that is

not a function.

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2. For the function 2 4g x x , do the following:

a. Represent the function verbally. b. Represent the function numerically (make a table of at least 5 values). c. Represent the function graphically (sketch the graph). d. State the domain and range using interval notation.

3. To determine the number of ounces of fluid that a person should drink in a day, divide his or her

weight in pounds by 2 and then 0.4 ounce for every minute of exercise. Write a function f that gives the fluid requirements for a person weighing 150 pounds and exercising x minutes a day. Evaluate and interpret 60f

4. A table of values is given. Determine whether the table represents a linear function or an

exponential function. Find a symbolic expression for the function.

a.

x -2 -1 0 1 2

f x 2

9

2

3

2 6 18

b.

x -2 -1 0 1 2

g x -10 -7 -4 -1 2

5. Write a system of three linear equations in three unknowns that represent the following situation and solve the system to answer the question: The largest angle in a triangle is 55° more than the smallest angle. The sum of the measures of the two smaller angles is 10° more than the measure of the largest angle. Find the measures of the three angles.

6. A golf ball is hit into the air, and its height h in feet after t seconds is given by 216 128h t t t .

a. After how many seconds does the golf ball reach its maximum height? b. What is the maximum height? c. When will the ball hit the ground?

7. In 2000 the population of Nevada was 2 million and growing continuously at an annual rate of

5.1%. The population of Nevada (in millions), x years after 2000 can be modeled by 0.051( ) 2 xf x e

( 0x corresponds to the year 2000)

a. Based on this model, what was the population in 2010? b. In what year does the model predict the population will reach 4 million (round to the nearest

whole number)?

math 55 intermediate algebra t....

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