unit 1:algebra review - scasd.org · cpa2 midterm review [name___date_pd___ cp algebra 2 -...

CPA2 Midterm Review [NAME______________________________DATE_______PD_____

CP Algebra 2 - Midterm Exam Review

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CPA2 Midterm Review MC and Free Response Practice

The midterm exam will have 2 parts: calculator allowed and calculator not allowed. Some questions will be multiple choice and some will be free response.

Unit 1:Algebra Review

Learning Targets: Topic A

Algebraic Expressions & Functions

1. I can simplify and evaluate algebraic expressions. 2. I can identify functions and evaluate functions written in function notation. 3. I can simplify and rationalize square roots.

Topic B Equations & Inequalities

4. I can solve equations, solve basic inequalities, and graph inequalities on a number line. 5. I can solve compound inequalities and absolute value inequalities. 6. I can graph linear functions. 7. I can graph and apply piecewise functions.

Topic D Writing

Equations

8. I can write equations of lines and piecewise functions. 9. I can apply equations of lines to real world problems. 10. I can use linear regression to solve real world problems.

Topic E Systems 11. I can solve systems of equations by substitution, elimination, and graphing.

Unit 2-1: Factoring and Solving Quadratics

Learning Targets:

0. I can add, subtract and multiply polynomial expressions

Factoring Quadratic

Expressions

1. I can factor using GCF. 2. I can factor by grouping. 3. I can factor when a is one. 4. I can factor when a is not equal to one. 5. I can factor perfect square trinomials. 6. I can factor using difference of squares.

Solving Quadratic Equations

7. I can solve by factoring. 8. I can solve by taking the square root. 9. I can perform operations with imaginary numbers. 10. I can solve by completing the square. 11. I can solve equations using the quadratic formula (with rationalized denominators). 12. I can use the discriminant to determine the number and type of solutions. 13. I can write quadratic equations given the real solutions.



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Unit 2-2: Writing and Graphing Quadratics Learning Targets:

Modeling with

Quadratic Functions

1. I can identify a function as quadratic given a table, equation, or graph. 2. I can determine the appropriate domain and range of a quadratic equation or event. 3. I can identify the minimum or maximum and zeros of a function with a calculator. 4. I can apply quadratic functions to model real-life situations, including quadratic

regression models from data.

Graphing

5. I can graph quadratic functions in standard form (using properties of quadratics). 6. I can graph quadratic functions in vertex form (using basic transformations). 7. I can identify key characteristics of quadratic functions including axis of symmetry,

vertex, min/max, y-intercept, x-intercepts, domain and range.

Writing Equations

of Quadratic Functions

8. I can rewrite quadratic equations from standard to vertex and vice versa. 9. I can write quadratic equations given a graph or given a vertex and a point (without a

calculator). 10. I can write quadratic expressions/functions/equations given the roots/zeros/x-

intercepts/solutions 11. I can write quadratic equations in vertex form by completing the square.

Unit 3: Polynomials & Polynomial Functions Learning Targets:

Polynomials: The Basics

1. I can classify polynomials by degree and number of terms. 2. I can use polynomial functions to model real life situations and make predictions 3. I can identify the characteristics of a polynomial function, such as the intervals of increase/decrease, intercepts, domain/range, relative minimum/maximum, and end behavior.

Factors and Zeros

4. I can write standard form polynomial equations in factored form and vice versa. 5. I can find the zeros (or x-intercepts or solutions) of a polynomial in factored form and identify the multiplicity of each zero. 6. I can write a polynomial function from its real roots.

Dividing Polynomials

7. I can use long division to divide polynomials. 8. I can use synthetic division to divide polynomials. 9. I can use synthetic division and the Remainder Theorem to evaluate polynomials.

Solving Polynomials

10. I can use the fundamental theorem of algebra to find the expected number of roots. 11. I can solve polynomials by graphing (with a calculator). 12. I can solve polynomials by factoring.

Finding and Using Roots

13. I can find all of the roots of a polynomial. 14. I can write a polynomial function from its complex roots.

Graphing 15. I can graph polynomials.



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Unit 1: Algebra Review (calculator allowed) 1. Evaluate the expression for the given value of the variable: 3x2 −2(x−6) for x =−2 A) 28 B) 4 C) -4 D) -28 E) none of these 2. Which one of the following relations is not a function? a) b) {(2, 3), (4, 6), (5, 6), (10, 9)} c) {(4, 0), (0, 4)} d) 3. Solve. a. Solve the equation: 5x−3(x+1)=9 A) x = 4 B) x = 3 C) x = 6 D) x = - 4 E) none of these b. Solve the inequality: 14x−3(x+8)≤9 A) x ≤−3 B) x ≤0 C) x ≤3 D) x ≥3 E) none of these c. Solve the absolute value inequality: 5 2x −3 < 35 (Free Response) ______________ 4. Use the function below to find 4 ⋅ f (−1) f (x)=−2x+7 A)36 B) 9 C) 20 D) 5 E) none of these

Domain: 1 3 8 0 2 3 5 Range: -3 4 8 1 0 2 -10

Domain: 1 2 3 4 5 Range: 0 2 0 2 0



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5. Evaluate using the provided piecewise function.

f (x )=2x 2 −5x +1,#if#x <−4−3x 2 −x , "if"−4≤ x <5x +2,#if#x ≥5

$

%&

'&

a) f (−6)= _________ b) f (−4)= _________ c) f (2)= _________ d) f (7)= _________ 6. Use the provided graph to answer each question.

a) Write a piecewise function for the provided graph.

f (x )=

!

"

###

$

###

b) Evaluate f (−1)= _________ c) Evaluate f (−5)= _________ d) Evaluate f (0)= _________

7. Solve using substitution or elimination. (Free Response) ______________

−3x+2y =17x−5y =−10⎧⎨⎩



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8. Simplify completely. Rationalize denominators if necessary.

a. 5 98 b. 528

c. 912 d.

30−15 36

e. 2 56

_______________ ______________ ______________ _______________ ______________

Unit 2: Quadratics (calculator allowed) 1. Write a quadratic equation in intercept/factored form with the following zeros. Then find the vertex of the equation.

a. 4 𝑎𝑛𝑑 −2 A. 𝑦 = 𝑥 + 4 𝑥 − 2 ; 1,−9 B. 𝑦 = 𝑥 + 4 𝑥 + 2 ; −9,1 C. 𝑦 = 𝑥 − 4 𝑥 + 2 ; 1,−9 D. 𝑦 = 𝑥 − 4 𝑥 + 2 ; −9,1

b. 6 𝑎𝑛𝑑 − 4 A. 𝑦 = 𝑥 + 6 𝑥 − 4 ; 1,25 B. 𝑦 = 𝑥 − 6 𝑥 + 4 ; 1,−25

C. 𝑦 = 𝑥 + 1 𝑥 − 24 ; 1,25 D. 𝑦 = 𝑥 − 6 𝑥 + 4 ; −25,1 2. A rectangle has a width of 𝑤 feet and a length of −4𝑤 + 50 . (Free Response)

a. Write an equation in standard form for the area of the rectangle. ______________

b. What width would maximize the area of the rectangle? What is the maximum area? Width______________ Area___________

3. Find the equation of the parabola in vertex form having the following vertex and y-intercept.

a. 2,−4 and a y-intercept of 4.

A. 𝑦 = 2𝑥! − 8𝑥 + 4 B. 𝑦 = 2 𝑥 − 2 ! − 4 C. 𝑦 = 2 𝑥 + 2 ! − 4 D. 𝑦 = 𝑥 − 2 ! − 4

b. 2,8 and a y-intercept of -4.

A. 𝑦 = −3 𝑥 − 2 ! + 8 B. 𝑦 = −3 𝑥 + 8 ! − 2 C. 𝑦 = −3𝑥! + 12𝑥 − 6 D. 𝑦 = −3𝑥! + 64



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4. Factor the following quadratic expressions completely. a. 4𝑥! − 36𝑥

A. 4 𝑥! − 9𝑥 B. 4𝑥 𝑥! − 9 C. 4𝑥 𝑥 + 3 𝑥 − 3 D. 𝑥 4𝑥! − 9

b. 3𝑥! − 12

A. 3 𝑥! − 4 B. 3𝑥 + 2 3𝑥 − 2 C. 3 𝑥 + 2 𝑥 − 2 D. 3𝑥 + 4 𝑥 − 3 c. 6𝑥! + 𝑥 − 35

A. 6𝑥 + 1 𝑥 − 7 B. 2𝑥 + 5 3𝑥 − 7 C. 𝑥 + 15 𝑥 − 14 D. None of these d. 2𝑥! + 11𝑥 + 15

A. 2𝑥 − 5 𝑥 − 3 B. 𝑥 + 3 2𝑥 + 5 C. 𝑥 + 5 𝑥 + 6 D. None of these

5. Rewrite the following equations in standard form.

a. 5 𝑥 − 2 ! + 4 A. 𝑦 = 25𝑥! − 100𝑥 + 104 B. 𝑦 = 25𝑥! − 20𝑥 + 8 C. 𝑦 = 25𝑥! + 8 D. 𝑦 = 5𝑥! − 20𝑥 + 24 b. 𝑦 = 2 𝑥 + 4 ! + 6

A. 𝑦 = 4𝑥! + 32𝑥 + 70 B. 𝑦 = 2𝑥! + 38

C. 𝑦 = 2𝑥! + 16𝑥 + 38 D. 𝑦 = 2𝑥! + 40𝑥 + 200 6. Solve the following quadratic equations. (Free Response)

a. 𝑥! + 5𝑥 + 6 = 0 b. 4𝑥! − 100 = 0 ______________ ______________



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6. Solve the following quadratic equations. (Free Response) c. 2𝑥! − 5𝑥 + 7 = 19 d. 3𝑥! − 294 = 0

______________ ______________

e. 4𝑥! − 26𝑥 − 14 = 0 ______________ 7. Solve each by completing the square. (Free Response)

a. 𝑥! − 16𝑥 + 72 = 0 b. 4𝑥! + 24𝑥 − 36 = 0 ______________ ______________ c. 2𝑥! + 16𝑥 − 104 = 0 d. 𝑥! + 28𝑥 − 81 = 0

______________ ______________ 8. Rewrite the equation in vertex form by completing the square. (Free Response)

a. 𝑦 = 𝑥! + 8𝑥 − 6 b. 𝑦 = 2𝑥! + 20𝑥 + 15 ______________ ______________ c. 𝑦 = 3𝑥! − 12𝑥 + 27 d. 𝑦 = 𝑥! − 9𝑥 + 4

______________ ______________



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9. Find the discriminant of each quadratic equation and give the number and type (real or complex) of solutions of each equation. (Free Response)

a. 5𝑥! − 6𝑥 + 10 = 0 b. 3𝑥! + 5𝑥 − 6 = 0 ______________ ______________

c . 2𝑥! − 7𝑥 + 12 = 0 d. 2𝑥! − 8𝑥 + 8 = 0 ______________ ______________

10. Factor completely: 2𝑥! + 11𝑥 + 15 (Free Response) ______________

Unit 3: Polynomials (calculator allowed) 1. What is the remainder when f(x) = x3 – 3x2 + x – 5 is divided by (x-2)?

a) 1 b) -3 c) -7 d) -11 e) 0 f) none of the above 2. The function f(x) = x3 + 6x2 + 11x + 6 has a zero at x = -1. Find the other two zeroes of the function.

a) x = -3, x = -2 b) x = -2, x = -1 c) x = -1, x = 2 d) x = 1, x = 3 e) x = 3, x = -3 f) none of the above

3. Using the graph of f(x) = (x-1)2+2, determine which of the following is TRUE.

a) The graph of f(x) has a relative MAXIMUM at x = 1 b) The graph of f(x) has a relative MINIMUM at x = 2 c) The function f(x) is increasing on the interval of x>1 or (1, ∞) d) The function f(x) is increasing on the interval of x<-1 or (-∞,-1) e) None of the above are true.



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4. Which of the following is a factor of f(x) = x3 – 10x2 + 32x – 32?

a) x b) x – 2 c) x + 2 d) x – 3 e) x + 4 f) none of the above 5. Write the polynomial function with zeros 3, 0, 5, and -2 in standard form. (Free Response) _______________________________________ 6. Write the polynomial function having rational coefficients with zeros -3 and 5+2i, in standard form. (Hint: remember conjugate pairs) (Free Response) _______________________________________ 7. Write the polynomial function having rational coefficients with zeros -3 and 2 11 in standard form. (Free Response) _______________________________________



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7. Which function has the graph to the right?

a. f(x) = 3x2 (x-1)(x-2)

b. f(x) = -x2(x-1)(x+2)

c. f(x) = x(x+12)(x-2) 2

d. f(x) = =x(x-1) 2 (x-2)

e. f(x) = -2x2 (x+1)(x-2) 8. For each function, state the total number of zeros, find all rational zeros, use other algebraic techniques to find all zeros, then list all zeros (simplified exact values). (Free Response) a. f(x) = x3– 5x2 + 8x – 6 b. f(x) = x3 – 9x2 + 25x – 25 Number of zeros: ____ Number of zeros: ____ Rational Zero(s) from calculator: _________ Rational Zero(s) from calculator: _________ List of ALL zeros: __________________ List of ALL zeros: __________________



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c. f(x)= x3 + x2 – 18x + 18 d. f(x) = x4+7x3+6x2-22x+8 Number of zeros: ____ Number of zeros: ____ Rational Zero(s) from calculator: _________ Rational Zero(s) from calculator: _________ List of ALL zeros: __________________ List of ALL zeros: __________________ e. f(x) = x3 + 3x2 +7x + 5 f. f(x) = x4 – 7x3 + 70x – 100 Number of zeros: ____ Number of zeros: ____ Rational Zero(s) from calculator: _________ Rational Zero(s) from calculator: _________ List of ALL zeros: __________________ List of ALL zeros: _________________



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Unit 2: Quadratics (NO CALCULATOR) 1. Graph using transformations. Clearly plot 5 “key” points. State the transformations in addition to the vertex, min/max and y-intercept. If there is not the transformation, write “NA” or “none”. (NC)

a. y = 3(x + 5)2 - 7 b. !!y = −12(x +2)

2 +7

Transformations: Reflection _________________ Vertical Stretch ______________ Left/Right __________________ Up/Down ___________________ Vertex ________________

Transformations: Reflection _________________ Vertical Stretch ______________ Left/Right __________________ Up/Down ___________________ Vertex ________________



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2. Graph. State if the parabola has a min or max, the vertex, equation of the axis of symmetry, y-intercept, and x-intercepts, domain and range. Clearly plot 5 points. (NC) a) !!y =2x

2 −4x +3 Min/Max of _______ at _________ (circle one) Vertex: ______________ Axis of Symmetry: _____________ y-intercept: _____________________ x-intercepts: ____________________ domain: ______________________ range: _______________________ b) y = -x2 - 8x -7 Min/Max of _______ at _________ (circle one) Vertex: ______________ Axis of Symmetry: _____________ y-intercept: _____________________ x-intercepts: ____________________ domain: ______________________ range: _______________________



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3. Simplify completely. (Free Response) a. −48 = ______________ b. −121 = ______________ c. −245 = ______________ d. −75 = _______________ e. 5 + 3𝑖 + 7 − 2𝑖 = _______ f. 5 − 2𝑖 − 7 − 2𝑖 = ______ g. 8 − 3𝑖 − 5𝑖 = _______________ h. 2 + 5𝑖 3 − 4𝑖 =__________ i. 3 − 4𝑖 5 − 6𝑖 = _________

Unit 3: Polynomials (NO CALCULATOR) 1. Fill in the missing information. Use interval notation as appropriate. A. For the given graph of: y = −2x4 − 2x3 + 5x2 + 5x + 4 Relative Minimum(s): __________________________________________ Relative Maximum(s): __________________________________________ Interval(s) of increase: __________________________________________ Interval(s) of decrease: _________________________________________ Intercepts: _________________________________ Domain: ________ Range: _________ End behaviors: sketch: B. Use the graph to the right. Relative Minimum(s): __________________________________________ Relative Maximum(s): __________________________________________ Interval(s) of increase: __________________________________________ Interval(s) of decrease: _________________________________________ Intercepts: _________________________________ Domain: ________ Range: _________ End behaviors: sketch:

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unit 1:algebra review - scasd.org · cpa2 midterm review [name_____date_____pd_____ cp algebra 2 -...

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unit 1:algebra review - scasd.org · cpa2 midterm review [name___date_pd___ cp algebra 2 -...