Name____________________________________ Math 2 Review

Unit 1 – Quadratics STANDARD: COMPLEX NUMBERS

Simplifying Radicals – √ √

√

√

√

Complex Numbers - √ and

Complex Conjugate – Switch the sign in front of the imaginary part.

Dividing Complex Number – Multiply by the conjugate of the denominator.

1. What is the value of ?

2. What is the value of ?

3. What is the product of ?

4. Find the difference of .

5. Write the conjugate of each of the following complex numbers.

a.

b.

c.

d.

6. Simplify the expression

.

7. Write each of the following expressions in standard

form.

√

√

√ √

√ √

8. What is the sum of ?

9. Find the product of .

10. Simplify the expression

.

11. Simplify each of the following expressions.

a. √

b. √

c. √

√

d. √

√

12. Simplify the expression .

STANDARD: ANALYZE QUADRATICS Domain – The x-values of the function, all real numbers for quadratics.

Range – The y-values of the function.

o Opens down: ] o Opens up:

Vertex – Max or min point of the graph.

Axis of Symmetry – The x-value that splits the graph in half, the x-value of the vertex.

Y-intercept – Y-value where graph intersects the y-axis.

X-intercept – X-value where graph intersects the x-axis.

Solutions – Also called zeros, roots, and x-intercepts.

Intervals of Increasing and Decreasing – Where the parabola is going up or down, split by the axis of symmetry.

Standard Form - ,

o vertex is

.

Vertex Form -

o vertex is . ** is opposite**

13. What are the domain, range, and solutions of the graph below?

14. Convert the following function to standard form.

.

15. Convert the following function to vertex form.

16. Determine the transformation of the following quadratics from the parent function .

a.

b.

c.

17. Use this graph of a quadratic function to answer the questions below.

a. Identify the vertex.

b. Identify the axis of symmetry.

c. Write an equation in the form:

, with

,

to represent this function.

d. What are the zeros and the y-intercept of the function?

e. What are the domain and range of the function?

f. On what interval is the function increasing?

g. On what interval is the function decreasing?

h. Convert the equation from part c to Standard Form.

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

STANDARD: SOLVE QUADRATICS Graphically - Simply look to see where the graph intersects the x-axis. Factoring - Use the “Big X”. Must be in standard form first! Find 2 numbers that multiply to be c and add to be b. These are called the factors. Set each factor equal to 0 and solve for x to find the solutions.

Quadratic Formula - Identify the a, b, and c values. Must be in standard form first! Plug the a, b, and c values into the quadratic formula.

o Simplify to solve for x.

o √

Discriminant - The discriminant is the number under the

radical of the quadratic formula, . Must be in standard form first!

o If the discriminant is positive, there are 2 real solutions.

o If the discriminant is 0, there is 1 real solution. o If the discriminant is negative, there are 2 imaginary

solutions. Quadratic Inequalities - Look at the inequality sign to determine if you need a solid or dotted line. If the inequality is or , use a dotted line. If the inequality is or , use a solid line.Test a point either inside or outside of the parabola to see if it is a solution. Shade the area of the graph that is the solution to the inequality.

18. Determine the types of roots for the graphed quadratics.

19. Find the value of the discriminant and explain the types of solutions for the quadratic .

20. Find the solutions of the equation .

21. Solve the equation by using the quadratic formula.

22. Sketch the function .

23. Determine whether the following points are a solution to the inequality .

a.

b.

c.

24. Determine the solutions to the graphs.

25. Find the solutions to the equation

.

Unit 2 – Right Triangle Trigonometry STANDARD: SPECIAL RIGHT TRIANGLES

Special Right Triangles –

26. Quadrilateral ABCD is an isosceles trapezoid with angles A and D measuring , segment BC

measuring , and segment CD measuring √ . What is the length of segment AD?

27. A square has a side length of 10 cm. What is the length of the diagonal of the square, rounded to the nearest whole number?

28. A 30 ft plank leaning against a building makes an angle of 60⁰ with the ground. How far from the base of the building is the plank?

29. The diagonal of a square is √ , what is the perimeter of the square?

30. What kind of triangles are formed when:

a. You construct the altitude of an equilateral triangle.

b. You construct a diagonal in a square?

31. Determine the lengths of the sides labeled and in the three diagrams below.

32. Find the length of the hypotenuse for the following triangles:

a. A right triangle that has legs with measures

of and √ .

b. A right triangle that has both legs with measures of 19.

33. A parallelogram has sides that are cm and cm

long. The measure of the acute angles of the parallelogram is . What is the area of the parallelogram?

34. An equilateral triangle has side lengths of 18. What is

the length of the attitude?

STANDARD: TRIGONOMETRIC RATIOS

Trig Ratios –

Sin =

Cos =

Tan =

Inverse Trig Ratios – Only used when finding the angle measure of a right triangle.

35. What does it mean for two angles to be complementary?

36. Angle and angle are complementary angles in a

right triangle. The value of is

. What is the

value of ?

37. Triangle is a right triangle with right angle , as shown. What is the area of triangle ?

38. A road ascends a hill at an angle of . For every 120

feet of road, how many feet does the road ascend?

39. Given triangle , what is ?

40. You are given that

. What is the measure

of angle ?

41. In a right triangle, if

, what is ?

42. In right triangle , if and are the acute

angles, and

, what is ?

43. Find the measure of angle . Round your answer to

the nearest degree.

44. Solve for .

45. Solve for .

46. A ladder is leaning against a house so that the top of the ladder is 18 feet above the ground. The angle

with the ground is 47. How far is the base of the ladder from the house?

2x

y 100

86

T

E

D

C

A

B

Unit 3 – Circles and Spheres STANDARD: CIRCLES

Area –

Circumference –

Parts of a Circle –

Properties of Tangent Lines – o Tangent and a radius form a right angle o You can use Pythagorean Theorem to find

the side lengths o Two tangents from a common external

point are congruent

Central Angles –

Inscribed Angles –

Angles Outside the Circle –

Intersecting Chords –

47. What is the value of in this diagram?

48. Given , with the inscribed quadrilateral, find the value of each variable.

49. is tangent to at point . measures 12

inches and measures 7 inches. What is the radius of the circle?

50. Given , the and the find the value of x.

51. If two tangents of meet at the

external point , find their congruent length.

52. The measure of is . What is the measure of ?

53. Isosceles triangle is inscribed in this circle.

and . What is the measure of ?

54. In this diagram, segment is tangent to circle at

point . The measure of minor arc is . What is

?

55. If and , find .

STANDARD: SPHERES

Surface Area –

Volume -

56. A sphere has a radius of 8 cm. What is the surface

area? Answer in both decimal and exact -form.

57. A sphere has a surface area of 50 m2. Find the

radius.

58. A soccer ball has a diameter of 10 in. Find its volume.

59. When comparing two different sized bouncy balls, by how much more is the volume of larger ball if its radius is 3 times larger than the smaller ball?

60. A hot air balloon is being deflated. Its full blown volume is about 80,000 ft

3.

After 30 minutes, the balloon’s radius

has decreased by

. What is the volume

of the balloon at this time?

61. A sphere has a volume of cubic inches. What is the surface area?

62. Find the volume of a hemisphere which

has a diameter of 7.

63. A sphere has a surface area of 81 m2. What must

its diameter be?

64. A balance ball has a surface area of 32 . What is its volume?

65. A sphere has a diameter of 32 cm.

a. What is the radius?

b. What is the circumference of the great circle?

c. What is the surface area?

d. What is the volume?

66. Find the volume of a hemisphere which has a radius of 5.

67. The radius of a sphere is changed from 20 cm to 5 cm. How has the surface area and volume of the sphere changes?

68. Find the diameter of a sphere which has a surface area of 200. Round to the nearest hundredth.

Class 1 Class 2

Mean 12 12

Median 12 11

Mode 10 and 11 13

Range 18 20

Standard Deviation 3.2 2.8

Class 1 Class 2

Mean 12 12

Median 12 11

Mode 10 and 11 13

Range 18 20

Standard Deviation 3.2 2.8

Winning 63 56 61 48 50

Losing 61 35 55 30 49

Unit 4 – Statistics

STANDARD: DATA ANALYSIS

Mean – Average of the numbers.

Median – Middle number when placed in order from smallest to largest.

Mode – Most repeated number.

Range – Difference between largest and smallest numbers.

Box Plot – Min, Q1, median, Q3, max.

Standard Deviation – Tells us how spread out the data is.

Normal Distribution -

Samples – Used when it is impossible collect data for an entire population. The means and standard deviations will vary from one sample to the next.

69. The winning margin for each game is the difference

between the winning score and the losing score. What is the mean and standard deviation of the winning margins for this data?

70. Compare the mean and standard deviations of the

two data sets.

Set A: 7, 3, 4, 9, 2 Set B: 5, 8, 7, 6, 4

71. The table below compares the age of students in Mrs. June’s 2 classes. Describe the similarities and differences. Which class has more variability in their age?

72. For a large population, the means is 4.8 and the standard deviation is 3.6. One random sample produces data values of 5, 1, 3, 4, 7, 6, 8, 2, 1, and 3. Another random sample produced data values of 8, 7, 5, 3, 4, 2, 2, 9, 7, and 3. Compare the means and standard deviations of the random samples to the population parameters.

73. Find the mean, median, mode, range, variance, and

standard deviation of the data set. If an outlier is defined as any value more than two standard deviations from the mean, which, if any, values in the data would be considered an outlier? 22, 18, 19, 25, 27, 21, 24, 18, 21, 21, 37

74. Describe a way in which a student could select people from their high school for a survey using least biased methods.

75. A data set describing the age of a population of 300 adults is normally distributed with a mean of 38 and a standard deviation of 4.

a. How many people would be expected to be older than 46?

b. How many people would be expected to be between 30 and 42?

Unit 5 – Piecewise, Exponential, and Inverses STANDARD: PIECEWISE FUNCTIONS Piecewise Function – A function made up of more than one part.

Point of Discontinuity – A break in the graph. It is identified by the x-value where the break occurs.

Step Function – A piecewise function made up of constant functions.

Absolute Value Functions – o | | o

Write Absolute Value as Piecewise – Set up your piecewise function as follows:

{

o Place a negative sign in front of the “a” value and then replace the absolute value symbol in the original function with parentheses and simplify. This is the first expression in the piecewise function.

o Replace the absolute value symbol in the original function with parentheses and simplify. This is the second expression in the piecewise function.

76. Evaluate the piecewise function:

{

a.

b.

c.

77. Determine the domain, range, and points of

discontinuity of the following step function.

{

78. Write the piecewise function for the given step function graph.

{

79. Solve the following absolute value equations.

a. | | b. | |

80. Identify the vertex and transformations of the following absolute value equations.

a. | |

b. | |

c.

| |

STANDARD: EXPONENTIAL FUNCTIONS

Properties of Exponents –

o Product of like bases:

o Quotient of like bases:

o Power to a power:

o Product to a power:

o Quotient to a power: (

)

o Zero exponent:

o Negative exponent:

or

Solving Exponential Equations – o Make the bases the same. o Set the exponents equal and solve for x.

Growth and Decay – o Growth: , Decay: o

Exponential Graphs – o Determine whether growth or decay o Horizontal Asymptote: o Y-intercept: Substitute 0 for

Translations – o Negative : reflects across x-axis o Negative : reflects across y-axis o Shifts horizontally by o Shifts vertically by

81. Solve the exponential equations.

a. b.

82. Jill invests $6000 at a rate of 4% interest compounded yearly. Approximately how much will Jill’s investment be worth in 5 years?

83. Mark buys a car new for $52,000. If the value of the car depreciates by 12% each year, how much will the car be worth in 4 years?

84. Identify a, b, h, k, domain, range, asymptote, y-intercept, end behavior, and translations. Then sketch the graphs.

(

)

85. A certain population changes according to the model

, where represents the time in years. What is the difference in the population between years 4 and 8?

86. Simplify the exponential expressions. a. b.

c.

d.

d. e.

STANDARD: INVERSES OF FUNCTIONS

Inverse of a Function – When x and y values switch. o Switch x and y variables. o Solve for y.

Composition – Verifies that two expressions are inverses of each other. Substitute one expression for x in the other expression, the resulting answer should be x if they are inverses.

Vertical Line Test – Determines whether a relation is a function. No more than 1 point of intersection.

One-to-One – There is exactly 1 y-value for each x-value (no x-values repeat and no y-values repeat). If a function is 1-to-1, then its inverse is also a function.

Horizontal Line Test – Determines whether a function is 1-to-1. No more than 1 point of intersection.

87. If , what is the domain and range of .

88. Find the inverse of the following functions.

a. b.

89. A table of ordered pairs for a function is shown.

a. Write the inverse table. b. Is a one-to-one function, why? c. Is the inverse table a function, why?

90. Using composition, verify that and

are inverse functions.

91. Using the graph of below, graph the inverse of the function. Is also a function?

92. Determine whether the functions below are one-to-one and determine whether the inverses of the graphed functions are also a function.

a.

b.

93. Using composition, and are inverses of

each other if and only if ] and ] equal

what?

Unit 6 – Algebraic Models for Quantitative Data STANDARD: ALGEBRAIC MODELS Scatterplot – A graph of plotted points that shows a relationship between two sets of data.

Types of Correlation – Positive, negative, or no correlation.

Correlation Coefficient – A value between and , notated by , that describes the strength of the correlation between two sets of data. Approximating Line of Best Fit – Draw a line through two points on the scatterplot to create a line that fits the data.

Find the slope:

.

Use point slope form: .

Regression – A linear, quadratic, or exponential equation that models the relationship between two variables. It is used to make predictions about data.

94. Create a scatterplot for the given data. Then

describe the correlation and approximate the line of best fit.

95. Describe the correlation for the following scatterplots.

96. Determine whether the following data would be best modeled by a linear, quadratic, or exponential equation?

97. Determine which of the following lines, , best describes the data displayed in the scatterplot, i.e., which would be considered a line of best fit?

98. Approximate the line of best fit for the given scatterplot.