Alg First Semester Final Review

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Semester final review

Multiple Choice

Identify the choice that best completes the statement or answers the question.

____ 1. A gem store sells beads made of amber and quartz. For 4 amber beads and 4 quartz beads, the cost is $46.0For 1 amber bead and 3 quartz beads, the cost is $14.50. Find the price of each type of bead.a. amber $10.00, quartz $1.50 c. amber $10.25, quartz $1.75 b. amber $9.75, quartz $1.25 d. amber $1.50, quartz $10.00

____ 2. Write 4 x3 + 8 x2 – 96 x in factored form.a. 6 x( x + 4)( x – 4) c. 4 x( x + 6)( x + 4) b. 4 x( x – 4)( x + 6) d. –4 x( x + 6)( x + 4)

____ 3. Find the zeros of . Then graph the equation.a. 3, 2, –3

2 4 6 –2 –4 –6 x

2

4

6

–2

–4

–6

y

c. 3, 2

2 4 6 –2 –4 –6 x

2

4

6

–2

–4

–6

y

b. 0, –3, –2

2 4 6 –2 –4 –6 x

2

4

6

–2

–4

–6

yd. 0, 3, 2

2 4 6 –2 –4 –6 x

2

4

6

–2

–4

–6

y

____ 4. The volume of a shipping box in cubic feet can be expressed as the polynomial . Eadimension of the box can be expressed as a linear expression with integer coefficients. Which expressioncould represent one of the three dimensions of the box?a. x + 6 c. 2 x + 3 b. x + 1 d. 2 x + 1



Factor the expression.

____ 5.a. c.

b. d.

____ 6. Solve . Find all complex roots.a.

−

7

5 ,

c.7

5 , b. no solution d.

−

7

5 ,

7

5

____ 7. Use the Rational Root Theorem to list all possible rational roots of the polynomial equation

. Do not find the actual roots.a. –4, –2, –1, 1, 2, 4 c. 1, 2, 4 b. no roots d. –4, –1, 1, 4

Short Answer

8. A science museum is going to put an outdoor restaurant along one wall of the museum. The restaurant spacwill be rectangular. Assume the museum would prefer to maximize the area for the restaurant.a. Suppose there is 120 feet of fencing available for the three sides that require fencing.

How long will the longest side of the restaurant be?b. What is the maximum area?

9. Use the graph of .a. If you translate the parabola to the right 2 units and down 7 units, what is the equation of

the new parabola in vertex form?

b. If you translate the original parabola to the left 2 units and up 7 units, what is theequation of the new parabola in vertex form?

c. How could you translate the new parabola in part (a) to get the new parabola in part (b)?

10. A local health official has determined that the function models the probability that arandomly chosen individual in the community will get the flu x days after the first reported case.a. Write the function in vertex form.b. How many days after the first reported case is the risk greatest that an individual will

become infected?

11. The diagram shows a storage building that consists of a cubic base and a pyramid-shaped top.a. Write an expression for the cube’s volume.b. Write an expression for the volume of the pyramid-shaped top.c. Write a polynomial expression to represent the total volume.



12. The volume in cubic feet of a workshop’s storage chest can be expressed as the product of its three

dimensions: . The depth is x + 1.a. Find linear expressions with integer coefficients for the other dimensions.b. If the depth of the chest is 6 feet, what are the other dimensions?

13. Graph the system of constraints. Then find the values of x and y that maximize .

Essay

14. Show that is equal to . Then use this to explain how you know that 5 is the

minimum value of the function.

15. A model for the height of a toy rocket shot from a platform is , where x is the time iseconds and y is the height in feet.a. Graph the function.b. Find the zeros of the function.c. What do the zeros represent? Are they realistic?d. About how high does the rocket fly before hitting the ground? Explain.

16. Find the rational roots of . Explain the process you use and show your work.

17. A manufacturing company’s profits are modeled by the equation , where y dollars is thtotal profit and x is the number of items manufactured. Graph the equation and explain what the x- and y-intercepts represent.

18. Write the equation of the line that contains the point (8, –3) and is perpendicular to . Graph thequation. Write the equation in standard form. Show your work.

19. A fish market buys tuna for $.50 per pound and spends $1.50 per pound to clean and package it. Salmon co$2.00 per pound to buy and $2.00 per pound to clean and package. The market makes $2.50 per pound proon tuna and $2.80 per pound profit for salmon. The market can spend only $106 per day to buy fish and $1 per day to clean it. How much of each type of fish should the market buy to maximize profit?a. Write an objective function P and constraints for a linear program to model the problem.b. Graph the constraint and find the coordinates of each vertex.c. Evaluate P at each vertex to find the maximum profit.

Other

20. Write a system of three equations that has a unique solution of (1, 2, 3).



Semester final review

Answer Section

MULTIPLE CHOICE

1. ANS: A PTS: 1 DIF: L3 REF: 4-7 Inverse Matrices and SystemOBJ: 4-7.1 Solving Systems of Equations Using Inverse MatricesTOP: 4-7 Example 4 KEY: word problem | systems and matrices | problem solving

2. ANS: B PTS: 1 DIF: L2 REF: 6-2 Polynomials and Linear FactOBJ: 6-2.1 The Factored Form of a Polynomial TOP: 6-2 Example 2KEY: factoring a polynomial | polynomial

3. ANS: D PTS: 1 DIF: L2 REF: 6-2 Polynomials and Linear FactOBJ: 6-2.2 Factors and Zeros of a Polynomial Function TOP: 6-2 Example 4KEY: Zero Product Property | polynomial function | zeros of a polynomial function | graphing

4. ANS: D PTS: 1 DIF: L2 REF: 6-3 Dividing PolynomialsOBJ: 6-3.2 Using Synthetic Division TOP: 6-3 Example 4KEY: division of polynomials | factoring a polynomial | polynomial | problem solving

5. ANS: B PTS: 1 DIF: L2 REF: 6-4 Solving Polynomial EquationOBJ: 6-4.2 Solving Equations by Factoring TOP: 6-4 Example 3KEY: factoring a polynomial | polynomial

6. ANS: A PTS: 1 DIF: L2 REF: 6-4 Solving Polynomial EquationOBJ: 6-4.2 Solving Equations by Factoring TOP: 6-4 Example 4KEY: factoring a polynomial | polynomial function

7. ANS: A PTS: 1 DIF: L2REF: 6-5 Theorems About Roots of Polynomial Equations OBJ: 6-5.1 The Rational Root TheoremTOP: 6-5 Example 1KEY: polynomial function | root of a function | solving equations | Rational Root Theorem

SHORT ANSWER

8. ANS:a. 40 ftb. 1,600

PTS: 1 DIF: L3 REF: 5-2 Properties of ParabolasOBJ: 5-2.2 Finding Maximum and Minimum Values TOP: 5-2 Example 3KEY: maximum value | quadratic function | area | problem solving | word problem | multi-part question

9. ANS:a.

b.

c. left 4 units, up 14 units

PTS: 1 DIF: L3 REF: 5-3 Translating ParabolasOBJ: 5-3.1 Using Vertex Form TOP: 5-3 Example 2KEY: parabola | translation | vertex form | problem solving | word problem | multi-part question

10. ANS:



a.

b. 8 days after the first reported case

PTS: 1 DIF: L3 REF: 5-7 Completing the SquareOBJ: 5-7.2 Rewriting a Function by Completing the Square TOP: 5-7 Example 7KEY: completing the square | quadratic function | multi-part question

11. ANS:

a.

b. or

c. or

PTS: 1 DIF: L4 REF: 6-1 Polynomial FunctionsOBJ: 6-1.2 Modeling Data with a Polynomial FunctionKEY: cubic function | modeling data | polynomial function | multi-part question

12. ANS:a. height, x – 1; width, x – 3b. height, 4 ft; width, 2 ft

PTS: 1 DIF: L3 REF: 6-3 Dividing PolynomialsOBJ: 6-3.2 Using Synthetic Division TOP: 6-3 Example 4KEY: division of polynomials | polynomial | problem solving | synthetic division | multi-part question

13. ANS:

0

0 1 2 3 4 5 6 7 8 9 x

1

2

3

4

5

6

7

8

9

10y

Vertices

(0, 0):(0, 2):

(3, 0):

(3, 5):

When x = 3 and y = 5, P has its maximum value of 270.

PTS: 1 DIF: L2 REF: 3-4 Linear Programming



OBJ: 3-4.1 Finding Maximum and Minimum Values TOP: 3-4 Example 1KEY: linear programming | maximum value | maximize

ESSAY

14. ANS:[4]

When , the value of is or 5. If x is any number other than

3, then and is a positive number. So will also be a positivenumber. The sum of a positive number and 5 has to be greater than 5. Therefore, the

value of is 5 when x is 3 and greater than 5 when x is any number other

than 3. Since is equal to , the minimum value of the

function is 5.[3] Most of the reasoning is correct, but one or two points of the argument were not

addressed thoroughly.[2] The reasoning was based too much on specific numerical values of x and y.[1] There were some correct observations, but there was no overall grasp of the situation.

PTS: 1 DIF: L4 REF: 5-2 Properties of ParabolasOBJ: 5-2.2 Finding Maximum and Minimum Values TOP: 5-2 Example 4KEY: quadratic function | minimum value | writing in math | reasoning | extended response | rubric-basedquestion

15. ANS:

[4] a.

2 4 6 8 10 –2 x

100

200

300

–100

–200

y

b. x

– 0



.05, x

9.11

c. Thezerosre

pr esentthetimesatwhi

chtheheightof thero

ck etis0.Theti



me – 0.05

secondsisnotrealisti

c.Thetime9.11sec

ondsisthetimeatwhi

chtherock etla



nds.

d. about3

36feet;Theheight

isthemaximumvalu

eof thefunction.

[3] an error in one of the three parts of the question[2] an error in two parts of the question[1] one part missing and errors in answer or reasoning for one of the other parts

PTS: 1 DIF: L3 REF: 6-2 Polynomials and Linear FactorsOBJ: 6-2.1 The Factored Form of a Polynomial TOP: 6-2 Example 3KEY: reasoning | graphing | graphing calculator | modeling data | polynomial function | problem solving |relative maximum | x-intercept | zeros of a polynomial function | extended response | rubric-based questionwriting in math | word problem

16. ANS:[4] Step 1:

List the possible rational roots by using the Rational Root Theorem. The leading



coefficient is 4 with factors of ±1, ±2, and ±4. The constant term is –1 with factors of –

1 and 1. The only possible roots of the equation have the form . Those

roots would be ±1, ± , and ± .

Step 2:Test each possible rational root in the equation. The only roots that satisfy the equation

are and 1.[3] an error in computation or missing part of the explanation[2] several errors in computation or in the explanation[1] one root given with no explanation

PTS: 1 DIF: L4 REF: 6-5 Theorems About Roots of Polynomial EquationsOBJ: 6-5.1 The Rational Root Theorem TOP: 6-5 Example 1KEY: extended response | polynomial function | Rational Root Theorem | root of a function | rubric-basedquestion | writing in math

17. ANS:[4]

O

40000 80000 120000 –40000 x

20000

40000

60000

–20000

–40000

–60000

y

The y-intercept represents the set-up costs and the x-intercept represents the leastnumber of items for which the company does not lose money, or a break-even point.

[3] minor errors in graph or explanation[2] correct graph with incorrect explanation or incorrect graph with correct explanation[1] no graph and errors in explanation or no explanation and errors in graph

PTS: 1 DIF: L4 REF: 2-2 Linear EquationsOBJ: 2-2.1 Graphing Linear Equations TOP: 2-2 Example 2KEY: x-intercept | y-intercept | graphing | linear equation | word problem | problem solving | extendedresponse | rubric-based question

18. ANS:[4]



The slope of the line perpendicular to this line is . Using the point-slope form of theequation, the line is

The standard form of this equation is or .

O 2 4 –2 –4 x

2

4

–2

–4

y

[3] one error in equation or graph

[2] two errors in equation or graph[1] correct answer but work not shown

PTS: 1 DIF: L4 REF: 2-2 Linear EquationsOBJ: 2-2.2 Writing Equations of Lines TOP: 2-2 Example 7KEY: standard form of linear equation | perpendicular | extended response | rubric-based question

19. ANS:[4] a. Let x be pounds of tuna and y be pounds of salmon.

The objective function is and the constraints are



b.

c.

The market should buy 28 pounds of tuna and 46 pounds of salmon to maximize profit.[3] two parts correct

[2] one part correct[1] correct answers, but no work shown

PTS: 1 DIF: L4 REF: 3-4 Linear ProgrammingOBJ: 3-4.2 Writing Linear Programs TOP: 3-4 Example 2KEY: maximize | objective function | vertices | word problem | problem solving | rubric-based question |extended response | maximum value | linear programming

OTHER

20. ANS:

Answers may vary. Sample:

PTS: 1 DIF: L3 REF: 4-7 Inverse Matrices and SystemsOBJ: 4-7.1 Solving Systems of Equations Using Inverse MatricesKEY: systems and matrices | unique solution | reasoning

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Alg First Semester Final Review