Name: __________________________Date: ____________ Period: ______

Advanced Algebra: Semester Exam Review (2011-2012)FINAL EXAM SCHEDULE:

January 23SEMESTER FINAL Exams:Period 4: 8:00am – 9:5 amPeriod 2 & A: 10:05 – 11:55 am

January 24SEMESTER FINAL Exams:Period 8: 8:00am – 9:5 amPeriod 7 & C: 10:05 – 11:55 am

January 25SEMESTER FINAL Exams:Per 1: 8:00am – 9:5 amPer 3 & D: 10:05 – 11:55 am

January 26SEMESTER FINAL Exams:Per 5: 8:00am – 9:5 amPer 6 & B: 10:05 – 11:55 am

Your Semester Exam will be comprised of two parts. The first part is a traditional multiple choice assessment similar to the quarter 1 assessment. The second part is an essay that will be your chance to display an in-depth understanding of a topic to be determined the date of the exam.

Essay Portion DescriptionYour final exam is devoted to writing about mathematics. This test will measure your ability to:

recognize major mathematical concepts, organize and express mathematical ideas clearly, develop and support the main idea, use appropriate math vocabulary and ability to define math terms in own words, use appropriate instruments and formulas to explain the concept and to solve problems ability to think critically using reasoning and evidence ability to apply the concepts and skills to solve application problems

Below is the list of 4 major mathematical concepts you will be tested on. The completion of the outlines for each concept is optional, but it is highly recommended to help you prepare for the exam. Remember the exam will be worth 10% of your overall grade in class (ONE LETTER GRADE). On the day of your test, you will have to explain in written form one math topic randomly selected at the beginning of the.

How to prepare for your testNow that you have the outlines, write an essay for each topic including example problems. Follow the outlines for each topic closely, you should know at least in principle what problems you will be working with. Prepare several example problems (1 is too few; 10 is too many) which you hope to explain while taking your test. Your essay is not graded on length, it will be graded on quality and clear demonstration of understanding of the concept presented.

In the essay portion of the test: 1.You must do some math. A discussion of the concept in words only is not appropriate. You need to show the knowledge of math behind your words. Ultimately, each topic depends on what you make of it. It is essential that your essay include some mathematics, not just words. 2. Your work must be original. This does not necessarily mean that you must do something nobody's ever thought of before, but you do need to work through the math yourself, and present the results in your own words, solve your own problems. 3. Readability. Your essay should be easy to read. Ask a friend to read it. Form a study group and read each other’s essays to prepare for the test. 4. Figures. By all means include lots of calculations and figures/ diagrams to demonstrate your understanding. You must describe each figure in the text in enough detail so the reader can figure out why it's there. 5. Length. Your essay should be 1-2 pages long not counting figures and lengthy calculations. Somewhat longer is OK; shorter is not. 6. Mathematical content. It's a good idea to get the math right! That’s why it is highly recommended you prepare your essays and study at least a couple of weeks before the final exam.

REVIEW SESSIONS:Exam Review sessions will be held instead of POW Review sessions in December and January. The purpose of the review sessions will be to individually review the work you have completed in

Semester Exam Review Days will be:Wednesday, December 14th from 1:15 to 2:15 p.m. in Room 321. Wednesday, January 11th from 1:15 to 2:15 p.m. in Room 321. Wednesday, January 18th from 3:15 to 4:15 p.m. in Room 321.

preparation for the final exam. You must come with your review packet and the work you have done! Do not wait until the last minute! Organize and prioritize! **NO HELP/ REVIEW SESSION WILL BE PROVIDED ON OR AFTER January 20th!!!!

Final Exam Essay Portion Scoring GuideYou will receive a letter grade for the essay portion according to the rubric below and the application problem will be graded based on a point system.Grade A (95%)A math essay in this category is outstanding, demonstrating clear and consistent understanding and mastery of mathematical concept, although it may have a few minor errors. Grade A also demonstrated high ability to apply the math skills to solve a real world application problem.

effectively and insightfully explains a math concept and demonstrates outstanding critical thinking, using clearly appropriate examples, reasons, and other evidence

is well organized and clearly focused, demonstrating clear coherence and smooth progression of ideas exhibits skillful use of math vocabulary Is free of major mathematical errors or misunderstandings application problem provides accurate solutions and supports in depth understanding

Grade B (85%)A math essay in this category is effective, demonstrating adequate understanding and mastery of mathematical concept, although it will have lapses in quality.

develops a point of view on the math concept and demonstrates competent critical thinking, using adequate examples, reasons, and other evidence

is generally organized and focused, demonstrating some coherence and understanding of math concept exhibits one major math error or accumulation of multiple minor math errors application problem demonstrates understanding but solutions include minor errors

Grade C (75%)A math essay in this category is inadequate, but demonstrates developing understanding and mastery of mathematical concept, and is marked by one or more of the following weaknesses:

develops a point of view on the math concept, demonstrating some critical thinking, but may do so inconsistently or use inadequate examples, reasons, or other evidence

is generally organized and focused, demonstrating some coherence and understanding of math concept is limited in its organization or focus, but may demonstrate some lapses in coherence or progression of ideas does not create example problems to demonstrate understanding of the concept uses weak math vocabulary or inappropriate word choice contains an accumulation of major mathematical errors application problem is complete, demonstrates weak understanding with incorrect solutions

Grade D (65%)A math essay in this category is seriously limited, demonstrating little or no mastery of mathematical concept, and is flawed by one or more of the following weaknesses:

develops a point of view on the math concept that is vague or seriously limited, demonstrating weak critical thinking, providing inappropriate or insufficient examples, reasons, or other evidence

is poorly organized and/or focused, or demonstrates serious problems with coherence or progression of ideas is limited in its organization or focus, but may demonstrate some lapses in coherence or progression of ideas does not create any example problems to demonstrate understanding of the concept uses weak math vocabulary or inappropriate word choice contains an accumulation of major mathematical errors demonstrates very little or no mastery of math concept displays fundamental errors and demonstrates severe flaws in mathematics application problem is incomplete and solutions are incorrect

Grade F (50% or less)A math essay in this category is not written, or does not address the prompt. Application problem is more than 50% incomplete and shows limited or no understanding of the concept.

OUTLINES for Essay Portion: 1. RIGHT TRIANGLES AND TRIGONOMETRY.

Pythagorean Theorem – define the right triangle, give the Pythagorean theorem, explain what it is and when is it used. Create and solve one example problem using the Pythagorean Theorem.

Pythagorean Triples – mention common combinations for right triangles Special Right triangles (30-60-90 and 45-45-90) provide a rule and an example for solving special right triangles Trigonometric ratios (sine, cosine, tangent) – write all three ratios, explain what they represent and when are they

used. Create and solve one example problem for each ratio. (you can use only one triangle to show application of all

three ratios!) Explain and show an example of using inverse trigonometric function to find the measure of an acute angle.

Law of Sine and Law of Cosine – write law of sine and law of cosine, explain when the law of sine and law of cosine are used and Create and solve one example problem.

2. EQUATIONS AND INEQUALITIES. Equations versus Inequalities – explain the difference between equations and inequalities as far as number of

solutions. Create and solve one example problem for a linear equation and inequality and quadratic equation and inequality. Describe the number of solutions graphically.

Solving systems of linear equations – explain different methods of solving systems of equations. (Graphing, elimination, substitution, matrices) Create and solve one example problem for each method, explain when is each method easier to use.

Solving system of linear inequalities – explain graphically how to identify solutions to a system of linear inequalities. Create and solve one example problem.

Solving non-linear systems - explain how to find solutions for non-linear systems of equations and inequalities. Create and solve one example problem. (for your example, keep it simple, use quadratic and linear equation or two quadratics, solve graphically)

3 FAMILIES OF FUNCTIONS. Functions – explain how would you determine if given graph or a table represents a function or not. Create one

example graph that is a function, and one example graph that is not a function. Create one table of values that is a function, and one example table of values that is not a function.

Function notation and Finding values of functions – explain how to find values of functions given an equation of the function. Create and solve at least two example problems. (Given f(x) = 2x+3 find f(-3) etc. )

Families of functions – list major families of functions and provide an equation and the graph for each family (Include: Linear, Power, Inverse power, Exponential, Absolute value, Trigonometric)

Domain and range – define domain and range of a function. Choose at least two different types of functions and identify the domain and range for each. Describe the difference between practical and theoretical domain and range (provide one example problem to describe the differences)

Piecewise functions –define piecewise function and show at least one example of a piecewise function. Identify the domain and range of your function.

4. QUADRATIC FUNCTIONS. Standard form of the quadratic equation – write the standard form of a quadratic equation, explain the quadratic,

linear, and constant term and their effect on the graph Factored form of the quadratic equation – write factored form of the quadratic equation, show one example of

factoring. Compare the benefits of having standard form of the equation to having factored form of the equation. Graphing quadratic equations - Explain graphing a quadratic function and finding zeroes, x-intercepts, maximum or

minimum, y-intercept, axis of symmetry. Create three different case of quadratic equations (based on the number of solutions) and graph each case.

Quadratic formula - Write and explain the quadratic formula. Show an example of using the quadratic formula to solve a quadratic equation, show how you can use the quadratic formula to find the vertex of the parabola, describe how the discriminant demonstrates number of solutions for the quadratic equation.

Review Problems for the Multiple Choice Section

The following are model problems to complete in preparing for the multiple choice section of the Advanced Algebra semester exam.

1. For each quadratic equation, find the roots, y-intercept and vertex then graph it:a. b. c. d.

2. Simplify.a. b. c.

d. e. f.

g. h. i. 3. Solve each equation (use the quadratic formula if necessary). Check your solutions by producing a graph on your

calculator.

a. b. c. d. e. f. g. h. i. x2 + 15x + 54 = 0 j. 2x2 + 5x – 3 = 0 k. x² + 2x + 9 =0 l. 2x2 – 11x + 15 = 0

4. Solve each System of Equations.

a. b. c. d. e.

5. Sketch a graph for the following inequalities. a. y < 2x + 3 b. c. y < x – 1 d. e. x + y > 2 f.

6. If the system of equations y = 2x – 5 and -3y = kx – 2 has no solutions, what is the value of k?

7. Solve the following literal equations for the indicated variable:

a. for b. for c

c. for d for r

8. If a rectangle measures 54 meters by 72 meters, what is the length, in meters, of the diagonal of the rectangle?

9. Given the quadratic equation f(x) = x² – 4x – 5 a. Sketch the graph and clearly identify the roots on the graphb. Find the coordinates of the vertex. Is it minimum or maximum? c. Solve the equation by factoring d. Find an equation of the axis of symmetrye. Use the quadratic formula to solve the equation.

10. The planning committee for the upcoming school play “Miss-terious” at LMSA asked the mathematics classes to give them some estimates about income that could be expected at different ticket price levels. The class did some market research to see what students would be willing to pay for tickets. They reported back the following model: I = -75p2 + 600p, where I stands for income and p for ticket price, both in dollars.

a. Find the predicted income if ticket prices are set at $3.b. Write equations that can be used to help answer each of the following questions. Then solve those equations,

check your solutions, and explain how you found the solutions.i. What ticket price will give income of $1,125?

ii. What ticket price will give income of $900?iii. What ticket price will give income of $970?

c. Find the price that will give maximum income, then find the maximum income.11. On its first day of business, the Great Mideastern Ice Cream Store sold two sizes of ice-cream cones, one scoop for

$1.00 and two scoops for $1.50. They sold 820 scoops of ice cream in cones for a total revenue of $690. At the end of the day, the manger wondered how many one-scoop and how many two-scoop cones they had sold, but no one had kept track. Represent the number of one-scoop cones sold by s and the number tow-scoop cones by t.

a. Write an equation relating s, t, and the number of scoops sold. (Note: equation will represent number of scoops sold, not number of ice creams!)

b. Write and equation relating s, t, and the total revenue from selling ice cream cones.c. Assuming that the store sold 50 one scoop cones and 213 two-scoop cones, what was the total revenue? d. Write an equation that expresses the number of one scoop cones s as a function of the two-scoop cones

t and the total revenue r.

12. If one leg of a right triangle is 8 inches long, and the other leg is 12 inches long, how many inches long is the triangle's hypotenuse?

13. In ABC, if A and B are acute angles, and sin A = , what is the value of cos A ?

14. In right triangle ABC to the right, what is the sine of A?

15. Lengths for the triangle below are given in feet. What is the measure of x?

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16. In the figure below, B is a right angle and the measure of A is 30°. If is 10 units long, then how many units long is ?

17. In right triangle ABC, tan A = 2.08.  Find mA to the nearest degree.

18. Find the area of the isosceles triangle STU.

19. Produce a graph of the following system of inequalities:

20. The Kepler Model Company is planning to market a variety of electric racing-car sets. Each set will contain at least 8 sections of curved track and 4 sections of straight track. No set will contain more than 36 sections in all or more than 20 sections of either type. If the company makes a profit of $0.40 on each straight section and $0.65 on each curved section, what combination of track sections will be most profitable for the company? (Hint: Write inequalities for the constraints and a separate equation to calculate profit…this is a Linear Programming Question!)

21. A certain publisher ships 300-450 books each week to a national chain of bookstores. Some of the books are shipped from the publisher’s eastern warehouse, and some are shipped from the publisher’s western warehouse, but at least one third of them must be shipped from each warehouse. The shipping cost per book is $0.37 from the eastern warehouse and $0.55 from the western warehouse. Find the minimum weekly shipping cost for these orders. (Hint: Write inequalities for the constraints and a separate equation to calculate profit.)

22. Find all missing sides and angles of the following triangles. a. b. c. d.

e. f.

23. Solve each system of equations

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a. b. c. d.

24. Solve for the variable a: -5a + 4(2 + 2a) = -125. Solve for the variable x: 3.2x – 1.7(x + 6)

26. Solve for the variable x:

27. Solve for the variable b:

28. In the figure below, , AB = 13 units, and BC = 10 units. What is the area and perimeter of ABC ?

29. Draw rhombus PLAN. Draw both diagonals in PLAN. If the perimeter of PLAN is 40 and PA = 16, find the length of diagonal LN.

30. For each equation below identify the family of functions this equation belongs to.

a. y = b. y = 3x(2.5 – ½ x) c. y = 2x³ - 10 d. y = e. y = 2½ x f.

31. For each of the following graphs, identify the family of functions to which it belongs.32.33.34.35.36.

32. Graph each piecewise function:

a. b. c.

33. Find the missing sides without using a calculator. Leave your answer in simplest radical form.

34. Evaluate the following compositions of functions. If f(x)=x+6 and g(x)=x2:a. (f○g)(a2) b. (g○g)(a+b) c. (f○f)(-6) d. (g○f)(-16) e. g(-5) f. (f+g)(x)g. (f – g)(x) h. f(g(x)) i. j. f(-12)

35. Find the equation of the line in slope-intercept form that passes through the point (-3, 5) and has slope of .

36. Find the equation of the line in slope-intercept form that passes through the point (0, -10) and is parallel to the line with equation .

37. Find the equation of the line in slope-intercept form that passes through the points, .

38. Convert into standard form.

39. Convert 5x + 10y = 15 into slope-intercept form.

45°

7 2

ba45°

45°

10c

d

45°

45°

45°

e

8 230°

60°

g

5 330°

8 2

h60°

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40. Determine whether either of the points (-1, -5) and (0, -2) is a solution to the system of equations, . Use

mathematics to explain your reasoning.

41. Solve the following nonlinear systems using algebra. Then, produce a graph for each system on graph paper that confirms your solution(s).

a. b.

42. In isosceles trapezoid ABCD shown to the right, and . Use the given information and what you know about line symmetry and angles of triangles to find the additional information requested below.

a. Find the measures of all angles shown in the diagram. b. Find the lengths of all segments shown in the diagram.

43. Recall that the perimeter of any rectangle can be expressed as a function of length L and width W of that rectangle in two different ways: P = 2L +2W or P = 2(W + L).

a. Find the perimeter of a rectangle that is 2.5 meters long and 1.2 meters wide. b. Find the length of a rectangle having a perimeter of 45 meters and width of 5 meters. c. Write an equation expressing length L as a function of width and perimeter of any rectangle. d. Write an equation expressing width W as a function of length and perimeter of any rectangle. e. Suppose you wished to determine the maximum area of a rectangular garden that could be enclosed with 100

meters of flexible fencing. i. Write an equation expressing area in terms of either length or width (using the perimeter equations

from above). ii. What is the best estimate of the maximum area and the dimensions of the enclosed rectangle?

44. Have you ever noticed that when you use a tire pump on a bicycle tire, the tire warms up as the air pressure inside increases? This illustrates a basic principle of science relating pressure P, volume V, and temperature T in a container.

For any specific system, the value of the expression remains the same even when the individual variables change.

a. For the expression to remain constant, what changes in pressure or volume (or both) must result when

the temperature increases? b. What changes in volume or temperature must result when the pressure increases? c. What changes in pressure or volume must result when the temperature decreases?

45. Planners of an amusement park estimate the number of daily customers will be related to the chosen admission price x (in dollars) by the function c(x) = 10,000 – 250x.

a. Calculate and explain the meaning of c(15) and c(30). b. Find the value of x satisfying the equation c(x)=4,000 and explain what it tells about the relation between

admission price and number of customers. c. Describe the practical domain and range of the function. d. Describe the theoretical domain and range of the function.

46. When Alicia and Jamal went to apply for restaurant jobs, they each found several different opportunities. Offer #1, Server: Pay is $7.50 per hour with work uniforms provided for free Offer #2, Server: Pay is $5.25 per hour and includes a $100 hiring bonus with the first week’s paycheck.

Uniforms again are provided for free. Offer #3, Host/Hostess: Pay is $8.75 per hour, but new clothes for this job cost about $250. The question for both of them was which offer to take. a. Write equations that will give the possible earnings under each plan as a function of the number of hours

worked. b. Produce graphs for all three relations for time worked from 0 to 250 hours. Explain how the graphs can be

used to find the best offer for various amounts of times worked. c. Produce tables showing the (hours worked, earnings) data for the three relations from 0 to 250 hours in steps

of 10 hours. Then explain how the entries help determine the best offer.d. Solve the following equation and inequalities. Explain what questions about the three offers can be answered

by the various solutions.

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47. When architects design buildings, they have to balance many factors. Construction and operating costs, strength, ease of use, and style of design are only a few. For example, when architects designing a large city office building began their design work, they had to deal with the following conditions:

The front of the building had to use windows of traditional style to fit in with the surrounding historic buildings. There had to be at least 80 traditional windows on the front of the building. Those windows each had an area of 20 square feet and glass that was 0.25 inches thick.

The back of the building was to use modern style windows that had an area of 35 square feet and glass that was 0.5 inches thick. There had to be at least 60 of those windows.

In order to provide as much natural lighting for the building as possible, the design had to use at least 150 windows.

a. Write the constraint inequalities for this situation and graph the feasible region. b. One way to rate the possible designs is by how well they insulate the building from the loss of heat in the

winter and the loss of air-conditioning in the summer. The heat loss R in Btu’s per hour through a glass

window can be estimated by the equation , where A stands for the area of the window in square

feet and t stands for thickness of the glass in inches. What are the heat flow rates of the traditional and modern windows? Use the results from above to write an objective function if the goal is to choose a combination of

traditional and modern windows to minimize the heat flow from the building. Find the combination of window types that meets the constraints and minimizes the objective

function. c. Minimizing construction cost is another consideration. The traditional windows cost $200 apiece and the

modern windows cost $250 apiece. Write an objective function if the goal is to minimize total costs. Find the combination of traditional and modern windows that will meet the constraints and

minimize total cost of the windows.

48. When a shoe company launches a new model, it has certain startup costs for the design and advertising. Then it has production costs for each pair of shoes that is made. When the planning department of Start Line Shoes estimated costs of a proposed new model bearing the name of a popular athlete, it reported that the average cost per pair of shoes

(in dollars) would depend on the number made, with the equation .

a. Calculate and explain the meaning of C(1), C(1,000,000) and C(2,500,000). b. For what value of x is C(x)=40, and what does this value tell about the business prospects of the new shoe

line? c. Sketch a graph of C(x) using 0 ≤ x ≤ 10,000,000 and 0 ≤ y ≤ 100. What type of function is represented in

this situation? d. What is the practical domain and range for this function? e. What is the theoretical domain and range for this function?

Good Luck and Happy Learning.

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