Algebra II Chapter 4 Test Review

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Algebra II – Chapter 4 Test Review Standards/ Goals:

(Algebra I): D.2.i.: I can transform different representations of relations and functions (specifically, with absolute value functions).

(Algebra II): E.2.b.: I can use transformations to draw the graph of a relation and then determine a relation that fits a graph.

A.1.c./A.SSE.2: I can take a quadratic expression and identify different ways to rewrite it. A.1.g./F.IF.7.b: I can graph a piecewise function. C.1.a./N.CN.1:

o I can understand that every complex number is written in the form of: a + bi. o I can recognize when a number is to be written as an imaginary number o I can use the conjugate when finding the quotient of complex numbers.

C.1.b./ N.CN.2.: I can add, subtract, and multiply complex numbers. C.1.c./N.CN.2.:

o I can simplify quotients of complex numbers. o I can factor a quadratic using complex conjugates. o I can find the imagery solution of a quadratic.

C.1.d./F.BF.1b.: o I can perform operations on functions, including: addition, subtraction, multiplication and division. o I can determine the domain and range of functions.

E.1.a./A.REI.4.b/F.IF.8a.: o I can solve a quadratic equation by completing the square. o I can solve an equation by find square roots. o I can solve a perfect square trinomial equation o I can use factoring and other methods to find the ‘zeros’ of a quadratic function. o I can solve quadratic equations using the quadratic formula.

E.1.a.: o I can solve a quadratic inequality. o I can identify the ‘zero’s’ of a quadratic function.

E.1.b./A.REI.4.b.: I can use the discriminant the number and type of roots for a given quadratic equation. E.1.d./A.CED.3.:

o I can solve quadratic systems graphically and algebraically with and without technology. o I can represent constraints by equations or inequalities and by systems of equations and/or

inequalities. o I can solve a system of linear OR quadratic equations by graphing. o I can solve a system of linear OR quadratic equations by using substitution.

E.2.a./F.IF.1: o I can understand what a relation and a function is. o I can understand that a function assigns to each element of a domain, EXACTLY one element of the

range. E.2.a./F.IF.2.: I can evaluate functions for input values in their domains. E.2.a./F.IF.5.: I can relate the domain of a function to its graph. E.2.a./F.BF.3.:

o I can determine the transformations that may occur with a quadratic function and decide whether it is a reflection, stretch, compression or a translation/shift and in what direction and by how many units

o I can identify the shape of a graph of a quadratic function. o I can identify both standard and vertex form of a quadratic function. o I can determine whether a quadratic function has a maximum or a minimum. o I can determine the domain and range of a quadratic function and graph the function with and

without technology. F.1.b.: I can find the zeros of a polynomial (specifically quadratics) in a variety of different ways. E.2.c.: I can solve a system of quadratic inequalities and can use the graph to determine a solution set.

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Find the domain and range of each of the following: #1. 𝑓(𝑥) = 5𝑥 − 9 #2. 𝑓(𝑥) = 𝑥2 − 6𝑥 + 10 #3. 𝑓(𝑥) = 5(𝑥 + 9)2 − 7 #4. Consider the following function: g(x) = −5|𝑥 + 9| − 4

a. Identify the more basic, parent function that has been shifted, reflected, stretched, or compressed.

Answer: y = _____ b. Indicate how the basic function found in step 1 has been shifted, reflected, stretched,

or compressed.

Horizontal shift: ᴑ LEFT ᴑ RIGHT ᴑ NONE # of units: ______

Stretch/Compress: ᴑ STRETCH ᴑ COMPRESS ᴑ NONE By a factor of: _____

X-Axis Reflection: ᴑ yes ᴑ no

Y-Axis Reflection: ᴑ yes ᴑ no

Vertical Shift: ᴑ UP ᴑ DOWN ᴑ NONE # of units: ______

#5. Consider the following function: g(x) = −𝟑|𝒙−𝟒|

𝟕+ 𝟓

a. Identify the more basic, parent function that has been shifted, reflected, stretched, or compressed.

Answer: y = _____ b. Indicate how the basic function found in step 1 has been shifted, reflected, stretched,

or compressed.

Horizontal shift: ᴑ LEFT ᴑ RIGHT ᴑ NONE # of units: ______

Stretch/Compress: ᴑ STRETCH ᴑ COMPRESS ᴑ NONE By a factor of: _____

X-Axis Reflection: ᴑ yes ᴑ no

Y-Axis Reflection: ᴑ yes ᴑ no

Vertical Shift: ᴑ UP ᴑ DOWN ᴑ NONE # of units: ______

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Consider the following: f(x) = |𝒙|. Write an equation of a new graph, if the following occur. #6. Reflected in the x-axis, translated/shifted 4 units left and translated/shifted up 6 units.

#7. Reflected in the x-axis, translated/shifted 6 units right and translated/shifted down 10 units.

#8. Reflected in the x-axis, translated/shifted 10 units left and translated/shifted down 8 units and stretched by a factor of 4.

#9. Reflected in the x and compressed by a factor of ½ and shifted down 12 units.

Consider the functions: 𝒇(𝒙) = 𝒙𝟐 + 𝟒 and g(x) = 3x – 5 #10. Find f(g(x)) and g(f(-6)) #11. Find f(-5) + g(5)

#12. Find f(x – 1) #13. Find g(x + 8) + f(-8).

Multiple Choice: #14. Evaluate the piecewise function at the given value of the independent variable.

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#15. Using the graph of the relation below, answer the following: i. Is the graph a function?

ii. Name the DOMAIN and RANGE of the relation in interval notation. Function:

ᴑ yes ᴑ no DOMAIN: ____________ RANGE: ______________







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#21. What would the vertex be of the following functions?:

a. 𝑦 = −9(𝑥 − 𝑏)2 − 𝑧 b. 𝑦 = 𝑡(𝑥 + 𝑤)2 C. 𝑦 = −3

4(𝑥 + 𝑟)2 + 𝑠

#22. Find the y-intercept of: 𝑓(𝑥) = −4(𝑥 − 1)2 − 10. Additionally, if this function is reflected across the y-axis, then what are the coordinates of the new vertex after this reflection has occurred?

POWER STANDARD: I can manipulate complex numbers (imaginary). Let m = 5 – 4i and h = 2 + 3i #23. Find m – 6h #24. Find m · h

#25. What is 𝑚

ℎ? #26. What is m + h?

Simplify each:

#27. √−36 #28. √−80

#29. (10i)(-12i) #30. (7 + √−100) − (−8 − √−121)

#31. (7 − 6𝑖)2 #32. (7 + 5i) + (3 – 8i)

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#33. 3i + -17 – 8i + 12 #34. 5 * 10i * -17i * - 9 #35. What is the complex conjugate of ¾ + 9i? #36. What is the complex conjugate of – 12i?

Suppose that g(x) = 7x + 9 and w(x) = 𝐱𝟐 − 𝟖 #37. Find (g + w)(-2) #38. Find (w – g)(-4)

POWER STANDARD: I can graph a quadratic equation. #39. Multiple choice: Which graph represents the function: 𝑦 = −𝑥2 + 2𝑥?

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#40. Multiple choice: Which graph represents the function? 𝑦 = −2𝑥2 + 4𝑥 + 2

POWER STANDARD: I can solve a quadratic equation. #41. Find the zeros of the following quadratic by using the quadratic formula: 𝑦 = 𝑥2 + 8𝑥 − 17

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#42. Solve the following: a. 𝑥2 − 11 = 89 b. 𝑥2 + 16 = 0 c. 𝑥2 − 55 = 9

#43. The factored form of a quadratic is given by: f(x) = (x – 9)(x + b). If f(x) has a y-intercept at (0, -27), the value of ‘b’ must be what?

POWER STANDARD: I can graph and solve a quadratic inequality. Solve each. Write in interval and set notation to represent the solutions. #44. 𝑥2 − 13𝑥 + 37 ≥ −5 #45. 𝑥2 + 16𝑥 − 7 < −35

POWER STANDARD: I can determine the number and types of roots of a quadratic equation using the discriminant. #46. Mark is working with the quadratic function 𝑦 = 𝑥2 − 10𝑥 + 7. Determine the number and type of roots for the equation using the discriminant.

#47. Consider the following quadratic: 𝑥2 + 8𝑥 = −6. Determine the discriminant of the quadratic and state the number and types of ‘roots’ that this quadratic will have.

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#48. Consider the parabolas shown below. What are some possible discriminant values for each.

POWER STANDARD: I can determine the domain and range of a quadratic function. #49. State the domain and range of the following parabolas.

PRACTICE MULTIPLE CHOICE: #1. C.1.b./N.CN.2.: What is the simplified form of (8 − 3𝑖)2?

a. 73 b. 16 – 6i c. 55 – 48i d. 55 + 48i

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#2. C.1.b./N.CN.2.: What is the simplified form of (-5i)(-3i)? a. -15i b. -15 c. 15 d. 15i

#3. E.2.a./F.BF.3.: Which of the following best describes how to transform 𝑦 = 𝑥2 to the graph of:

𝑦 = 4(𝑥 − 2.5)2 − 3? a. Translate 2.5 units left, stretch by a factor of 4, translate 3 units down. b. Translate 3 units right and 2.5 units down, stretch by a factor of 4. c. Translate 2.5 units right, stretch by a factor of 4, translate 3 units down. d. Stretch by a factor of 4, translate 2.5 units left and 3 units down.

#4. C.1.b./N.CN.2.: Simplify: (5 + 6i) + (2 – 3i).

a. 7 + 3i b. 3 + 3i c. 4 d. 7 – 3i

#5. E.2.a./ F.IF.8a.: What is the maximum value of the function 𝑦 = −3𝑥2 + 12𝑥 − 8?

a. 4 b. -8 c. 8 d. 2

#6. A.APR.3 The quadratic function f(x) = −2(𝑥 + 3)2 − 7 has a y-intercept of:

a. (0, -25) b. (0, -7) c. (0, -14) d. (0, -18)

#7. C.1.b./N.CN.2.: What is the complex conjugate of: ½ - 2i?

a. 2 – 2i b. 2 – ½ i c. ½ i – 2 d. ½ + 2i

#8. E.2.a.: Which function has the same range as 𝑦 = (𝑥 + 3)2?

a. 𝑦 = (𝑥 + 3)2 − 2 b. 𝑦 = 𝑥2 + 9 c. 𝑦 = 2(𝑥 − 3)2 + 1 d. 𝑦 = (𝑥 − 5)2

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#9. E.1.c./N.CN.7.: Solve: 𝑥2 + 6𝑥 + 18 using the quadratic formula. a. No solution b. 0, -6 c. -3 ± 3i

d. -3 ± 3√3

#10. E.1.d.: Solve 𝑥2 + 2 = 6 by graphing the related function.

#11. E.1.a.: What is the solution set for 5𝑡2 + 6 = 8𝑡?

a. {−3

5±

𝑖

5√31}

b. {4

5±

2

5√14}

c. {−4

5±

𝑖

5√14}

d. {4

5±

𝑖

5√14}

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#12. E.1.a.: What are the solutions for 𝑥2 = −4𝑥 + 7?

a. −2 ± √11

b. −4 ± √23 c. −7, 1

d. 2 ± √3 #13. E.2.a.: Which equation is the reflection of 𝑦 = 𝑥2 − 4𝑥 + 3 across the x-axis?

a. 𝑦 = 𝑥2 − 4𝑥 + 3 b. 𝑦 = 𝑥2 − 4𝑥 − 3 c. 𝑦 = −𝑥2 + 4𝑥 − 3 d. 𝑦 = −𝑥2 + 4𝑥 + 3

#14. E.2.a.: Andrew threw a spear into the air. The function = ℎ(𝑡) = −5𝑡2 + 30𝑡 + 5 can be used to determine the height, ‘h’ of the spear after ‘t’ seconds. What is the maximum height that the spear reached?

a. 30 feet b. 50 feet c. 60 feet d. 90 feet e. 120 feet

#15. MULTIPLE CHOICE: What is the product of (4 – 3i) and (-7 – 2i)?

a. -23 + 13i b. -23 – 29i c. -34 + 13i d. -34 – 29i

#16. MULTIPLE CHOICE: Rationalize 1+𝑖

1−𝑖

a. -1 b. 1 c. –i d. i

#17. E.2.a.: The function 𝑦 = (𝑥 + 5)2 + 7 is reflected across the y-axis. What are the coordinates of the vertex after this reflection?

a. (-5, -7) b. (-5, 7) c. (5, -7) d. (5, 7)

#18. C.1.d./F.BF.3.: What is the parent function for f(x) =−5(𝑥 − 9)2 − 10?

a. f(x) = x b. 𝑓(𝑥) = 𝑥2 c. 𝑓(𝑥) = |𝑥|

d. f(x) = √𝑥

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PRACTICE FREE RESPONSE QUESTION: Consider the following quadratic:

𝑓(𝑥) = 4𝑥2 + 20𝑥 + 10 a. What are the coordinates of the vertex? Axis of symmetry? Min/max and where at? Y-

intercept?

Consider the following quadratic: 𝑓(𝑥) = 4𝑥2 + 20𝑥 + 𝑐

b. For what value(s) for ‘c’ will the quadratic have TWO REAL solutions?

c. For what value(s) for ‘c’ will the quadratic have ONE REAL solution?

d. For what value(s) for ‘c’ will the quadratic have TWO COMPLEX/NO REAL solutions?

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Algebra II Chapter 4 Test Review