math 2 final exam skills review - lexington public...

Math 2: Algebra 2, Geometry and Statistics

Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434

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MMath 2 Final Exam – Skills Review

Each of the following is a discrete mathematical skill, followed by practice problems on that skill. You should be able to complete each of these problems, but you should also be able to apply each one to solutions of other, more complex, problems. Unit 1 – Function Concept Write an explicit and a recursive function rule for a linear table of values.

1. Fill in the difference column; then write a recursive and an explicit rule for the functions represented by each of the tables below.

a.

x f(x) ∆ 0 9 1 11 2 13 3 15 4 17

b. x f(x) ∆ 0 3 1 6 2 9 3 12 4 15

Recursive rule:

𝑓 𝑥 =

, 𝑥 = 0

, 𝑥 > 0

Recursive rule:

Explicit Rule: f(x) =

Explicit Rule:



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Describe in words and write a recursive function rule for a non-‐linear table of values.

2. Answer the following questions and describe the pattern in words

Describe whether a relation is a function based on an equation, a table, or a graph.

3. Determine whether each rule describes a function or merely a relation. Explain your reasoning.

a. 𝑦! = 2𝑥 − 1

b. 𝑦 = 3𝑥! − 2

a. x f(x) ∆ 0 -‐2 1 -‐1 2 2 3 7 4 14

b. x f(x) ∆ 0 1

1 3 2 9 3 27 4 81

What is the second difference? What order polynomial does this table represent? Rule in words:

Recursive Rule

Rule in words:



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4. Determine whether each graph represents a function or merely a relation. Explain your reasoning.

a.

b.

c.

5. Determine whether each table represents a function or merely a relation. Explain

your reasoning. a.

x y 2 8 5 7 2 6 7 5 9 4

b. x y 1 6 2 5 3 6 4 4 5 7

4

2

–2

–4

–6

–5 5



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Find domain and range of a function based on an equation, a graph, and a table.

6. Find the domain and range of each function. Use interval or inequality notation.

a. 𝑓 𝑥 = 3𝑥! − 6𝑥 + 9 Domain: Range:

b. 𝑓 𝑥 = 𝑥 + 4 Domain: Range:

c. 𝑓 𝑥 = !!!!

Domain: Range:

7. Find the domain and range of each function shown below. The entire function is

shown. a.

Domain: Range:

b.

Domain: Range:

8. Determine the domain and range of the function represented by the table below.

The table represents the complete function. x 5 7 8 6 y 3 10 12 1

Domain: Range:

8

6

4

2

–2

–5 5

4

2

–2

–4

–6

–5 5



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Evaluate functions, operations with functions, and compositions of functions.

9. Given the functions 𝑓 𝑥 = 2𝑥! + 1 and 𝑔 𝑥 = 4𝑥 − 3, find each value.

a. 𝑓(−2)

b. 𝑔(−2) c. 𝑓(−2) ∙ 𝑔(−2)

d. 𝑓(𝑔 −2 )

e. 𝑔(𝑓 −2 )

f. 𝑔(𝑓(4)

10. Given 𝑎 𝑥 = 4𝑥 + 3, 𝑏 𝑥 = 𝑥! + 5, and 𝑐 𝑥 = −𝑥 + 6, find a formula for each

composition.

a. 𝑎(𝑏 𝑥 )

b. 𝑏 𝑐 𝑥

c. 𝑎(𝑐 𝑥 )

d. 𝑐(𝑎 𝑥 )



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Find the inverses of functions given as tables, graphs, and equations.

11. Find the inverse of each function.

a. 𝑓 𝑥 = 5𝑥 − 1

b. 𝑔 𝑥 = −3𝑥 + 9

12. Find and draw the inverse of each of the functions. Then determine whether the

inverse is a function. a.

Is the inverse a function?

b.

Is the inverse a function?

4

2

–2

–4

–6

–5 5

4

2

–2

–4

–6

–5 5



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Unit 2 – Statistics Distinguish between categorical and non-‐categorical data.

13. Determine whether each type of data is best treated as categorical or non-‐categorical. Explain your answers.

a. Number of ticket sales at Red Sox games.

b. Red Sox players’ uniform numbers.

c. Height of homerun balls.

d. Number of homeruns hit by a player in a season.

e. Homeroom number.

f. Type of soda purchased.

g. Preference of hot drink.

h. Seat number at Fenway Park.



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Create a two-‐way table from a list of data.

14. Use the following list of data to fill in the two way table.

Name Gender Favorite Color George Male Blue Rachel Female Red Matthew Male Blue Lauren Male Green Alex Male Yellow Christine Female Yellow Andrea Female Blue Dea Female Blue Jonah Male Purple Susan Female Purple Jeff Male Blue Jocelyn Female Blue Kenyon Female Beige Anna Female Blue Adina Female Brown Leiran Male Blue Lev Male Grey Simon Male Green Bob Male Blue Janet Female Red

Gender Male Female Total

Favorite Color

Blue

Not Blue

Total



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15. Using the table above, complete the following table showing conditional probabilities for color preference.

Gender Male Female Total

Favorite Color

Blue 1.0

Not Blue 1.0

Total 1.0

16. Choose 3 values from the table. For each one, write a complete sentence explaining what it means.

17. Create a segmented bar graph for the conditional probabilities that you just

calculated.

18. Is having blue as your favorite color associated with being male? Use your table and/or your bar graph to explain your answer.



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The graph below shows the population and the number of area codes for each state in the United States.

19.

a. Draw in an appropriate line of best fit on the scatterplot. b. Write the equation of the best-fit line in point-slope form.

c. Change the equation from part b into slope-intercept form.



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20. Use the scatterplot from the previous page to answer these questions. a. Does this data appear to be correlated? How can you tell? Use an approximation

of the r-‐value in your answer.

b. Suppose that a state has 10 area codes. How big do you suppose its population would be? Explain how you got your answer.

c. When this data was collected, the population of Canada was approximately 29.1 million. If a similar pattern exists there, how many area codes do you think Canada has (or had)? Explain how you got your answer.

d. When this data was collected, Great Britain had a population of approximately 58.6 million. If Great Britain used the same system, how many area codes do you think it would need? Do you think this estimate would be better or worse than the one for Canada? Explain your reasoning.



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Unit 3 – Coordinate Geometry Use tools of coordinate geometry to find the midpoint of a segment, find the distance between two points, find the slope of a segment or line, and determine whether two lines are parallel, perpendicular, or neither.

21. Quadrilateral ABCD has coordinates A (-‐12, 0), B (-‐5, 4), C (3, 2), and D (-‐4, -‐2). Find the length, slope, and midpoint of each side of the quadrilateral.

Length Slope Midpoint Segment AB

Segment BC

Segment CD

Segment DA

22. Use the information above to classify ABCD as a parallelogram, rhombus, rectangle,

square or simply a quadrilateral.



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Find the equation of a line based on a description.

23. Find the equation for a line that has x-‐values that are twice as large as its y-‐values.

24. A circle has an equation (𝑦 − 4)! + (𝑥 + 2)! = 4. Find the equations of two vertical tangents and two horizontal tangents to this circle.

25. (Refer to the circle and tangent lines above.) Find the equation for a line that passes through the bottom-‐most point of tangency and the right-‐most point of tangency.

26. Find the equation for a line that has a slope of 3 that passes through the point (4, 7)

27. Find the equation for a line that is perpendicular to the line 3x – y = 10 and that passes through the point (-‐4, 5).



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Find the intersection of two lines.

28. Find the point where each pair of lines below intersects. Use an efficient method.

a. 430−−=

=+−

xyyx

b. 93102

=−

=+

yxyx

Determine whether a triangle is acute, right, or obtuse based on side lengths

29. Each set of lengths below represents possible lengths for three sides of a triangle.

ii. Determine whether the lengths could make a triangle. iii. If a triangle is possible, determine whether it is isosceles, equilateral, or scalene. iv. If a triangle is possible, determine whether it is acute, obtuse, or right.

a. 3 cm, 4 cm, 7 cm

b. 5 miles, 5 miles, 3 miles



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c. 2 in, 2 in, 2 in

d. 10 m, 22 m, 25 m

e. 5 ft, 12 ft, 13 ft

f. 7 yd, 24 yd, 25 yd

Recognize the basic graphs including: 𝑦 = 𝑥, 𝑦 = 𝑥!,𝑦 = 𝑥!,𝑦 = 𝑥 , 𝑥! + 𝑦! = 1, 𝑦 =𝑥, and 𝑦 = !

!.

30. Sketch each of the parent graphs below. You should be able to do this from

memory.

𝑦 = 𝑥

𝑦 = 𝑥!

𝑦 = 𝑥! 𝑦 = 𝑥 ,



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𝑥! + 𝑦! = 1

𝑦 = 𝑥

𝑦 = !!

𝑦 = 𝑥! − 𝑥

Translate basic graphs up down, left and right and determine the resultant equation. Reflect basic graphs over the x-‐axis or the y-‐axis. Stretch or shrink graphs by a given factor. Determine how any of the preceding transformations changes the equation of the graph.

31. Sketch the graph each of the following equations. Name the parent function and describe in words how the parent function has been transformed.

a. y−1 = (x −3)2

Describe the transformation(s):

2y = −x

b.


c. (x −1)2 +(3y)2 = 16

Describe the transformation(s)

d. 𝑦 = !!!


e.

€

2y = x − 5


f. −y = x −2( )3−(x −2)




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32. Each of the following figures is one of the basic graphs with one or more transformation performed on it. For each one,

i. Identify the parent function. ii. Describe in writing the transformation(s) that was/were performed. iii. Write an equation for the new function.

a.

i. ii. iii.

b.

i. ii. iii.

c.

i. ii. iii.

d.

i. ii. iii.

8

6

4

2

–2

–4

–6

–8

–10

–10 –5 5 10

8

6

4

2

–2

–4

–6

–8

–10

–10 –5 5 10

8

6

4

2

–2

–4

–6

–8

–10

–10 –5 5 10

8

6

4

2

–2

–4

–6

–8

–10

–10 –5 5 10



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33. Write an equation for each of the

following transformations of 𝑦 = 𝑥

a. The graph is shifted left 3 units.

b. The graph is dilated horizontally by a factor of 3.

c. The graph is reflected over the x-‐axis.

d. The graph is shifted down 5 units.

e. The graph is transformed such that its range is [−4,∞).

f. The graph is transformed such that the y-‐intercept is (0, 3).

34. Write an equation for each of the following transformation of 𝑦 = 𝑥!.

a. The graph is shifted up 3 units. b. The graph is reflected over the y-‐axis.

c. The graph is reflected over the x-‐axis and shifted left 3 units.

d. The graph is dilated horizontally by a factor of ½.

e. The graph is transformed such that its domain is [3,∞).

f. The graph is shifted left 2 units and up 3 units.



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Find lines of reflection, and degrees and centers of rotations given a preimage and an image.

35. Describe in detail the transformation that was performed to map the figure onto the image.

a.

b.

c.

d.

e.

f.



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Unit 4 – Deductive Geometry Identify the pairs of congruent and supplementary angles formed when lines intersect, when parallel lines are cut by a transversal, and the interior and exterior angles of polygons.

36. Find the measures of missing angles.

37. Determine whether each statement is sometimes (S), always (A) or never (N) true

based on the figure.

1. ∠8 ≅ ∠9 2. ∠1 ≅ ∠2 3. ∠4 𝑖𝑠 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑡𝑜 ∠5 4. ∠7 ≅ ∠4 5. ∠1 𝑖𝑠 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑡𝑜 ∠8 6. ∠1 𝑖𝑠 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑡𝑜 ∠4 7. ∠7 𝑖𝑠 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑡𝑜 ∠6

w

vt

s

r

q

A

B

a = b = c = d = e =

f = g = h = k = n =

p = q = r = s = t =

v = w =

98

7

6

5 4

3

2

1



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38. For each of the figures below a) Identify any congruent triangles and write a congruence statement. If

there are no congruent triangles write CBD (for cannot be determined). b) Write the congruence shortcut used. If there are no congruent triangles,

explain why not. i.

ii.

iii.

iv.

v.

O is the center of the circle. (Hint: Remember what you know about the definition of a circle.)

vi.

D

A C

B



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vii.

viii.

ix.

39. Complete each of the following proofs. Use a separate sheet of paper.

a.

b.

c.

G

FE

D

H

I

JG H

H

K

L

J



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40. Draw a figure and write a proof for each of the following statements. a. b.



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Unit 5 – Exponents Simplify radicals.

41. For each equation, find the value of k that satisfies the equation.

a. 98 = 𝑘 2

b. 125 = 𝑘 5

c. 5+ 125 = 𝑘 5

d. 2 6+ 150 = 𝑘 6

42. Simplify as much as possible. Leave no negative exponents. Simplify numerical

exponents when the exponent is 4 or less.

a. 𝑥! ! !! b. 𝑥!" ∗ !!!

! c. 𝑥! ∗ 𝑥! !

d. 3𝑥!𝑦! ! e. 3 ! ∗ (5)! f. !!!!!

!!!!

43. Write each radical in simplified or standard form.

a. 32

b. 121 c. 45

d. !!

e. !! f.

!! !



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Simplify exponents, including fractional exponents.

44. Simplify as much as possible. Leave no negative exponents.

a. 4!! ⋆ 4

!!

b. 25!!!

c. 27!! ⋆ 27

!!

d. 9!!!

e. 8!!!!!

Determine whether numbers are rational or irrational.

45. Determine whether each of the following numbers is rational or irrational. Explain your answer.

a. 16.33

b. 52− 3

c. !.!"#!

d. 13



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Unit 6 – Polynomials and Quadratics Factor Polynomial expressions including

• Finding common factors. • Difference of perfect squares. • Monic and non-‐monic quadratics using trial and error, sums and differences of

roots, Sasha’s method (the good, the bad, and the ugly).

46. Factor the following polynomial expressions completely.

a.

b. c.

d.

e. f.

g.

h. i.



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Solve polynomial equations by factoring and using the zero product property.

47. Factor each equation. Then use the zero product property (ZPP) to solve for x.

a. = 0

b. = 0

c. = 0

d. = 0

Use vocabulary to describe polynomials.

48. Write a polynomial that meets each description.

a. Has a degree of 4 and a quadratic term with a coefficient of 2.

b. A binomial with a degree of 3 and no linear term.



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Add, subtract, and multiply polynomials.

49. Let 𝑔 𝑥 = 3𝑥! − 2𝑥! + 5.

a. Find ℎ(𝑥) such that 𝑔 𝑥 + ℎ 𝑥 = 4𝑥! − 3𝑥

b. Find 𝑗(𝑥) such that ℎ 𝑥 − 𝑗 𝑥 = −𝑥! − 2𝑥! − 1

c. Find 𝑘 𝑥 such that 𝑔 𝑥 ∗ 𝑘 𝑥 has a degree of 6 and 𝑔 𝑥 + 𝑘 𝑥 has degree 1 and no quadratic term.

Solve quadratic equations using an efficient method. Hint: the value of the discriminant of a quadratic equation (𝑏! − 4𝑎𝑐) tells you several things about the solutions. If the discriminant is… Then…

Positive The quadratic has two solutions

Zero The quadratic has one real solution

Negative The quadratic has no real solutions

A perfect square The quadratic is factorable

Not a perfect square The quadratic is not factorable



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50. Solve the following quadratic equations. Use the most efficient method.

a.

b. −3𝑥! + 24𝑥 − 52 = 0 c. 4 = 4(𝑥 − 3)!

d. e.

f.



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Find equations of quadratic functions based on various given information.

51. Write a quadratic equation for each of the following sets of parameters. a. A parabola with zeros at − !

! and 3 that

passes through (4, 27)

b. A parabola with x-‐intercepts of – 3 and 4 and a y-‐intercept of – 24.

c. A parabola with a vertex at (3, 5) passing through (1, 17).

d. A parabola with a vertex at (2, 7) that passes through (3, 3).



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Free Multi-Width Graph Paper from http://incompetech.com/graphpaper/multiwidth/


Graph quadratic equations from standard, factored, and vertex forms.

52. Graph each of the following equations. Clearly label each of the features. Consider your scale when graphing. a. 𝑦 = −3(𝑥 − 4)! + 3

Clearly label the following:

The vertex:

The zeros:

The axis of symmetry:

The y –intercept:

A mirror/sister point:

b. 𝑦 = 2(𝑥 − 3)(2𝑥 + 3)


The vertex:

The zeros:


The y –intercept:




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c. 𝑦 = 6𝑥! − 13𝑥 − 5


The vertex:

The zeros:


The y –intercept:


Show the equivalence of multiple forms of solving a quadratic equation.

53. Given the equation 𝑦 = 2𝑥! − 9𝑥 − 5

a. Solve by completing the square

b. Solve using the quadratic formula.



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Unit 7 – Probability

Solve problems about probability experiments. 54. Solve each of the following problems. Give your answer as a calculation (using

factorial and/or choose notation) and as a number.

a. If you flip a coin 6 times, what is the probability that you will get exactly 4 heads?

b. If you flip a coin 5 times, what is the probability that you will get at least 3 heads?

c. A spinner has four equal sections labeled 1, 2, 4, and 8. If you spin it twice, what is the probability that the product of your numbers is 8?

d. A bag contains 3 orange, 5 pink, and 2 green marbles. If you draw two marbles in a row without replacement, what is the probability that both marbles are the same color?

e. Refer to the situation in part d. If you draw three marbles in a row without replacement, what is the probability that both marbles are the same color?



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Use two-‐way tables to determine independence and conditional probability. The table below shows polling data about whether students complete homework regularly and whether they are on honor roll. Use the table to answer the questions that follow.

Always Completes Homework

(A)

Sometimes Completes Homework

(S)

Never Completes Homework

(N)

Total

On Honor Roll (H) 290 421 15 726 Not on Honor Roll (R) 589 229 656 1474

Total 879 650 671 2200

55. Calculate each of the following probabilities and explain in writing what it means.

a. P(A) = Explain:

b. P(N) = Explain:

c. P(A and N) = Explain:

d. P(A or N) = Explain:

e. P(H|R) = Explain:

f. P(R|N) Explain:

g. P(A|H) = Explain:

h.



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56. Is being on honor roll independent of always doing homework? Explain what

calculation(s) supports your conclusion.

57. Is sometimes completing your homework independent of being on honor roll? Explain what calculation(s) supports your conclusion.



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Unit 8 – Similarity and Trigonometry Use the properties of similarity to determine lengths and angles between two similar figures.

58. Find the measures of the missing sides and angles. a.

b.

59. Decided whether the figures are similar. Explain why or why not. a.

b.



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Draw scaled figures using the parallel or ratio methods.

60. Draw a dilation of the figure with a scale factor of !!, then answer the following

questions.

a. What is the ratio of the perimeter of the

dilated quadrilateral to the perimeter of the original figure?

b. What is the ratio of the area of the dilated quadrilateral to the area of the original figure?

Determine if two triangles are similar using triangle similarity shortcuts.

61. Write a similarity statement (e.g. ∆𝐴𝐵𝐶 ∼ ∆𝐷𝐸𝐹) and a similarity shortcut for each figure.

a.

b.



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62. Prove that ∆𝐷𝐴𝐺 ∼ ∆𝐶𝐴𝑇. Use the properties of similarity to find the lengths, areas and volumes of similar figures and solids.

63. Find the missing measures in each similar triangle. a.



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64. Find the missing measures in each pair of similar figures. a.

b.

c.

d.



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Use trigonometric ratios to solve problems.

65. Write a true trigonometric equation and solve for the missing measure. Round to the nearest tenth.

a.

b.

c.

d.

e.

f.



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66. Find the missing measures. Give an exact answer. a.

b.

c.

67. Find the area of each figure.

a.

b. ABCD is a rectangle.



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Solve problems using the properties of circles.

68. Find the area of the shaded portion of each figure and the length of each intercepted arc. Give an exact answer (in terms of pi).

a.

b.

69. Find the missing measure.

a.

b.

70. Find the missing measures.

a.

b.

math 2 final exam skills review - lexington public...

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