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Advanced Algebra: Semester Exam Review (2011-2012)

FINAL EXAM SCHEDULE:

|January 24 |January 25 |January 26 |January 27 |

|SEMESTER FINAL Exams: |SEMESTER FINAL Exams: |SEMESTER FINAL Exams: |SEMESTER FINAL Exams: |

|Period 4: 8:00am – 9:5 am |Period 8: 8:00am – 9:5 am |Per 1: 8:00am – 9:5 am |Per 5: 8:00am – 9:5 am |

|Period 2 & A: 10:05 – 11:55 am |Period 7 & C: 10:05 – 11:55 am |Per 3 & D: 10:05 – 11:55 am |Per 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 will have short response in which work must be shown or explanations of understanding must be given in writing.

REVIEW SESSIONS:

Exam Review sessions will be held instead of POW Review sessions in January. The purpose of the review sessions will be to individually review the work you have completed in 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!!!!

What should you be able to do for the final? Use the outline below as a checklist of concepts we have covered this semester.

1. RIGHT TRIANGLES AND TRIGONOMETRY.

• Pythagorean Theorem – define the right triangle, give the Pythagorean theorem, explain what it is and when is it used.

• 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. 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.

2. EQUATIONS AND INEQUALITIES.

• Equations versus Inequalities – explain the difference between equations and inequalities as far as number of solutions. Create and solve problems 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) Solve problems 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.

• Solving non-linear systems - explain how to find solutions for non-linear systems of equations and inequalities.

• Linear programming- write constraints and objective functions, find feasible region and vertices, find vertex that satisfies the objective

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. 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 cases 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.

Practice Review Problems

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. [pic] b. [pic] c. [pic] d. [pic]

2. Simplify.

a. [pic] b. [pic] c. [pic]

d. [pic] e. [pic] f. [pic]

g. [pic] h. [pic] i. [pic]

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

a. [pic] b. [pic] c. [pic] d. [pic]

e. [pic] f. [pic] g. [pic] h. [pic]

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. [pic] b. [pic] c. [pic] d. [pic] e. [pic]

5. Sketch a graph for the following inequalities.

a. y < 2x + 3 b. [pic] c. y < x – 1

d. [pic] e. x + y > 2 f. [pic]

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. [pic] for [pic] b. [pic] for c

c. [pic] for [pic] d [pic] 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 graph

b. 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 symmetry

e. 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 [pic]ABC, if [pic]A and [pic]B are acute angles, and sin A = [pic], what is the value of cos A ?

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

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

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

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

18. Find the area of the isosceles triangle STU.

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

[pic]

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!)

20. 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.

[pic] [pic] [pic][pic]

e. f.

[pic] [pic]

23. Solve each system of equations

a. [pic] b. [pic] c. [pic] d. [pic]

24. Solve for the variable a: -5a + 4(2 + 2a) = -1

25. Solve for the variable x: 3.2x – 1.7(x + 6)

26. Solve for the variable x: [pic]

27. Solve for the variable b: [pic]

28. In the figure below, [pic] , AB = 13 units, and BC = 10 units. What is the area and perimeter of [pic]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 = [pic] b. y = 3x(2.5 – ½ x) c. y = 2x³ - 10 d. y = [pic] e. y = 2(½ x( f. [pic]

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. [pic] b. [pic] c. [pic]

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

34. Evaluate the following functions. If f(x)=x+6 and g(x)=x2: a. g(-5) b. (g)(x+2)

c. f(g(x)) d. f(y – 3) e. f(-12)

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

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 [pic].

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

38. Convert [pic]into standard form.

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

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

42. In isosceles trapezoid ABCD shown to the right, [pic] and [pic]. 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 [pic]remains the same even when the individual variables change.

a. For the expression [pic]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.

• [pic]

• [pic]

• [pic]

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 [pic], 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 [pic].

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?

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45°

[pic]

[pic]

[pic]

45°

[pic]

[pic]

45°

45°

45°

[pic]

[pic]

[pic]

45°

[pic]

[pic]

60°

30°

60°

[pic]

[pic]

30°

[pic]

*3 ................
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