Algebra II Name: Page 1 of 16

Study Guide: Midterm Exam The purpose of this study guide is to HELP you to prepare for the final exam. It does not include EVERY piece of information that you should know. You are responsible for ALL material covered during the semester, all information from the class notes and all homework problems assigned. 1. Vocabulary that you should know:

Variable Numerical expression Variable expression Evaluate Absolute value Opposites Reciprocals Simplify Term Like or Similar Term Coefficient Combining Equation in one variable Solution Solution set Root Solve No solution Empty set Null set

€

∅ Real numbers

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ℜ Formula Origin Integer

Graph Coordinate Greater Than Less than Conjunction Disjunction Combined Inequality Equation with two variables Ordered pair Linear Equation Linear Parallel Perpendicular Quadrant x-intercept y-intercept Point Distance formula Midpoint Midpoint formula Point-Slope formula System of Linear Equations Substitution Elimination Pascal’s Triangle

Answer the following. All answers are in class notes from this semester. 2. Know the order of operations. 3. Properties that you should know.

• Commutative • Associative • Identity • Inverse • Property of Opposites • Property of Reciprocals • Distributive • Multiplicative Property of 0 • Identity Property of Multiplication • Multiplicative Property of -1

Algebra II Name: Page 2 of 16 4. Know the THREE ways to write division.

5. Know the difference between

€

−x 2 and

€

−x( ) 2.

6. Know the golden rule of algebra. 7. Know: What does it mean if you are solving an equation or inequality and all of the variables go

away? (No solutions, many solutions) 8. Know how to translate phrases into variable expressions. (Ex: sum, more than, increased by, less,

less than, of, product, difference, quotient, decreased by, opposite, reciprocal, twice, half, etc.) 9. Know how to graph inequality on a number line. (open circle vs. closed circle) 10. Know what multiplying or dividing both sides of an inequality does to the inequality sign. 11. Know the difference between conjunctions and disjunctions and how to graph them on a number

line. 12. Know how to solve the story problems like those that we covered in Chapters 1 and 2. 13. Know the strategy to get rid of fractions when solving equations and inequalities. 14. How many solutions does an absolute value equation have? There are three answers to this. 15. How do you find the solution(s) for absolute value inequalities? There are four answers to this. 16. Know the Standard form and Slope-Intercept form of equations for a line and how to rewrite a

linear equation from one form to the other. 17. Know how to graph a line given the equation or given a point and the slope.

Algebra II Name: Page 3 of 16 18. Know how to identify the x and y intercepts from either the equation for the line or from the graph

of the line. 19. For horizontal lines, know the slope, how to graph and how to write an equation for the line. 20. For vertical lines, know the slope, how to graph and how to write an equation for the line. 21. Know how to find the slope of a line if given:

• The graph • Two points on the line • The equation in standard form • The equation in slope-intercept form

22. Know how the slopes of parallel and perpendicular lines are related. 23. Know how to find the equation of a line whether given two points on the line or one point on that

line and the slope of that line. 24. Know how to solve a system of equations using substitution or elimination. 25. Know how to solve a system of three equations with three variables. 26. Know how to divide polynomials using long division. 27. Know how to find the range of a function when given the domain of that function.

28. Know how to find the domain of a function. What two conditions limit or exclude values from the

domain?

29. Know how to find a linear function f(x) when given m, b or any function value.

30. Know the difference between

€

−x 2 and

€

−x( ) 2.

31. Know how to ADD and SUBTRACT polynomials by combining similar terms. 32. Know the Laws of Exponents and how to use them.

33. Know how to multiply binomials using F-O-I-L.

Algebra II Name: Page 4 of 16 34. Know how to expand powers of binomials.

Algebra II Name: Page 5 of 16 Practice Problems Simplify each expression completely. 1.

€

− −5 2.

€

2 −5

3.

€

5 − −2 4.

€

−5 − 2

Use <, = or > to make a true statement.

5.

€

3 ____ − 3 6.

€

3 2 • 4 2 ____ 3 • 4( )2

7.

€

9 − 3( ) − 2 ____ 9 − 3 − 2( )

Simplify completely.

8.

€

4 2 − 6 ÷ 2 + 3 9.

€

4 2 − 6# $ % &

' ( ÷ 2 + 3

10.

€

4 2 − 6# $ % &

' ( ÷ 2 + 3( ) 11.

€

14 − 2 9 − 2 5 − 3( )[ ]

12.

€

64 ÷ 4 2 + 3 3 2 − 1# $ % &

' ( 13.

€

2 2 3 2 + 4 2" # $ %

& '

10 2

Evaluate each expression if

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x = 3 , y = 2 and

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z = 5 .

14.

€

2 x 2 + x − 2 15.

€

xz − yz( )3

16.

€

4 xyz

z 2 − x 2 17.

€

z 2 − y 2 − x 2

xy

#

$ % %

&

' ( (

5

Algebra II Name: Page 6 of 16 18.

€

x 2 − z 2

Evaluate each expression if

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x = 7 and

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y = 3 .

19.

€

3 x 2 − y 2# $ % &

' ( + 5 x − y( )

2

x + y( )2

Simplify each expression completely.

20.

€

2

3x

"

# $ $

%

& ' '

3

2y

"

# $ $

%

& ' ' 21.

€

1

31 • 3( ) + −3 + 3( )

22.

€

1

3• 3

"

# $ $

%

& ' ' x + −1( )[ ] + 1 23.

€

17 − −26( )

24.

€

−15 − −40( ) 25.

€

−9.6 − 13.4

26.

€

−87 + 16 − 22 + 61 27.

€

−16 − 2 − −61( )[ ] − 2 + −6( )[ ] 28.

€

−22 − 33 − 16 − 7 29.

€

7 4 x − 5y( )

30.

€

2

36 x − 9y − 27( ) 31.

€

6 m − 4 n + −7 m( ) − −5n( ) 32.

€

4 3 − y( ) + 2 1 − y( ) 33.

€

2 −5 x( ) −6y( )

34.

€

−a( ) −b( ) + 6 ab( ) −1( ) 35.

€

4 2 y 2 − 3# $ % &

' ( − 2 4 + y 2#

$ % &

' (

Algebra II Name: Page 7 of 16 Simplify each expression completely.

36.

€

−c d + 5( ) + 6 2 − cd( ) 37.

€

−5( )3

−1

5

#

$ % %

&

' ( (

2

38.

€

24 ÷ −3( )4 −5( ) ÷ 2

39.

€

60 ÷ −5( )[ ] 8 ÷ −2( )[ ]

40.

€

24 − 6 x 2

2 41.

€

−12( ) −3

4−

1

2

#

$ % %

&

' ( (

5

9÷ −10( )

42.

€

−1

2÷

1

4

#

$ % %

&

' ( ( ÷ −4( ) 43.

€

−3( ) −4( ) −2( )−6( ) −2( )

Solve each equation.

44.

€

4 x + 11 = 3 45.

€

15 −x

8= −1

46.

€

2

5x − 2( ) = x + 4 47.

€

3 1 − x( ) + 5 = 3 1 + x( ) − 7

48.

€

3y − 2 y − 1( )6

= −1 49.

€

47 =4

5x + 7

Algebra II Name: Page 8 of 16 Solve each equation.

50.

€

2 x

3−

7 − x

4= 1 51.

€

8 2 x − 3( ) = 4 4 x − 8( )

52.

€

6 x + 9 = 3 2 x + 3( ) 53.

€

7 − 2 x

5= −1

Solve each equation for the given variable. 54.

€

2 x − 5y = 10 , x 55.

€

y = mx + b , x 56.

€

C = 2 π r , r 57.

€

P = 2 + 2w , w Translate each phrase into a variable expression. 58. A number decreased by 2 _______________ 59. Five more than three times a number _______________ 60. The difference between a number and its square _________________ 61. The sum of twice a number and six _______________ 62. Twice the sum of a number and six _______________ Solve the problem. 63. The perimeter of a rectangular garden is 200 feet. If the length of the garden is 10 feet longer

than its width, what are the length and width of the garden? Solve each inequality and graph its solution set.

64.

€

3 x + 16 < 7 65.

€

11 −3

2x > −4

Algebra II Name: Page 9 of 16 Solve each inequality and graph its solution set. 66.

€

5 3 − x( ) < 7 − x 67.

€

3t > 6t + 12

68.

€

2

3x − 2 − 3 x( ) < 5 x + 2 1 − x( ) 69.

€

5 2 x + 3( ) > 2 x − 3( ) + x

70.

€

5 + X < x − 8 71.

€

5 x + 3 < 5 x + 10

72.

€

4 x > 2 3 + 2 x( ) 73.

€

2 x < −1

2−6 − 4 x( )

74.

€

x − 7 − x( ) > 2 x + 1 75.

€

4 x + 3 2 − 3 x( ) < 5 2 − x( )

76.

€

x > −2 and

€

x < 3 77.

€

−4 >3

4x − 7 > −7

78.

€

x < 0 and

€

x ≥ 4 79.

€

1 − x > 4 and

€

x − 1 < 4 80.

€

−2 ≤ 3 − x < 4 81.

€

x + 7 < 4 or

€

7 − x < 1

Algebra II Name: Page 10 of 16 Solve each equation or inequality and graph the solution set. 82.

€

x ≥ −2 or

€

x ≤ 3 83.

€

5 x − 1 > 0 and

€

4 x + 2 < 0

84.

€

2 x + 3 > 1 or

€

5 x − 9 ≤ 6 85.

€

−2 < 1 −x

2≤ 3

86.

€

−1 ≤ 3 x + 2 ≤ 8 87.

€

x ≥ −1 and

€

x ≥ 3 88.

€

x ≥ −1 or

€

x ≥ 3 89.

€

x = 2

90.

€

2 x + 3 = 5 91.

€

x > 2

92.

€

x < 2 93.

€

x > −2

94.

€

x < −2 95.

€

x > 0

96.

€

2 x − 5 = 0 97.

€

3 x + 2 > 4

98.

€

4 x − 7 ≥ 0 99.

€

x − 2

4≤

1

2

Algebra II Name: Page 11 of 16 Solve each equation or inequality and graph the solution. 100.

€

2 x − 1 + 3 ≤ 6 101.

€

4 − 3 x + 1 < 2

102.

€

3 x + 5 ≤ −8

103. Complete each ordered pair to form a solution for the equation

€

4 x + 3y = 8 .

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0 , ___( ) , 5 , ___( ) , ___, 0( ) 104. Graph the line with the equation

€

2 x − y = 6.

105. Graph the line with the equation

€

y = −1

3x + 1 .

106. Give the coordinates for the y-intercept for the line with the equation

€

x + 2 y = 4 . 107. Find the slope of this line:

€

2 x + 4 y = 5

108. Find the slope of this line:

€

y =2

5x − 7

109. Find the slope of the line containing the points

€

6 , −5( ) and

€

−4 , 3( ).

Algebra II Name: Page 12 of 16 Find the equation of a line with the given information. Then rewrite your equation in standard form.

110. Having slope of

€

1

5 and passing through the point

€

4 , −3( ) 111. Containing the points

€

1 , −4( ) and

€

4 , −5( )

112. Having slope of

€

3

2 and having y-intercept of

€

5

113. Being PARALLEL to

€

x + y = 5 and passing through the point

€

0 , 3( ) 114. Being PERPENDICULAR to

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x − 3y = −2 and passing through the point

€

−1 , 2( ) 115. What is the slope of a horizontal line? 116. What is the slope of the y axis? 117. What is the slope intercept form of an equation of a line? 118. What is the point-slope formula and what is it used for? 119. What is the standard form of an equation of a line? Graph the following linear inequalities. 120.

€

3 x + 2 y ≥ −4 121.

€

x + 2 y ≤ 2

Algebra II Name: Page 13 of 16

122.

€

2 x − 3y < 6 123.

€

y < 3 x + 3

y >1

3x + 1

"

# $

% $

124. Find the distance between the points

€

5 , 3( ) and

€

1 , 4( ). 125. Find the midpoint of the line segment with endpoints

€

−3 , −5( ) and

€

−1 , 6( ) .

Solve each system of equations.

126.

€

2 x − 7y = 105 x − 6y = 2

# $ %

127.

€

x = 3y + 22 x − 5y = 2

# $ %

128.

€

y = 3 x − 9y = −x − 1

# $ %

129.

€

3 x − y = 9y = x + 1

# $ %

130.

€

2 x + 7y = −75 x + 7y = 14

# $ %

131.

€

6 x − 4 y = 59 x − 6y = 5

# $ %

132.

€

2 x − 2 y = 8y = x − 4

# $ %

133.

€

2 x − y = 74 x − 2 y = 9

# $ %

Algebra II Name: Page 14 of 16

134.

€

4 x + y − 3 z = 112 x − 3y + 2 z = 9

x + y + z = −3

#

$ %

& %

135.

€

2 x + y + 3 z = 9x − 2 y + z = 8

−4 x + 3y + 2 z = −4

#

$ %

& %

If

€

f x( ) = 3 − x 2 and

€

g x( ) = x + 2 , find the following.

136.

€

f g 1( )( ) 137.

€

f 3( ) + g g 5( )( )

138. f g x( )( ) 139. f x + 1( )

140. Given

€

f x( ) = 2 x + 1 with domain

€

D = −2 , −1 , 0 , 1 , 2{ } , find the range of f.

State the domain of each function.

141.

€

f x( ) =2 x + 1

x + 3( ) x − 4( ) 142.

€

f x( ) =x + 1

x

143. Given a linear function f , what is the significance of the value of

€

f 0( )?

Algebra II Name: Page 15 of 16 Find an equation of the linear function f using the given information.

144.

€

m = −2 , b = 5 145.

€

m =3

2, f 3( ) = 7

146.

€

f 2( ) = −5 , f 4( ) = −2 147.

€

m = −1 , f 0( ) = −3

148. If f is a linear function where

€

f 1( ) = −1 and

€

f 4( ) = 8 , find

€

f 0( ) and

€

f −6( ) .

Simplify completely.

149.

€

4 a 2 + 3ab − b 2# $ % &

' ( − b 2 − 2 ab#

$ % &

' ( 150.

€

2 n 2 − n + 5# $ % &

' ( + n 2 + 1#

$ % &

' (

151.

€

3 x 2 −2 xy 2# $ % &

' (

2 152.

€

−ab 4# $ % &

' (

2

153.

€

4 x 3y" # $ %

& '

3−3 xy 4" # $ %

& '

2 154.

€

t 2 ⋅ t n +1 ⋅ t n −3

Multiply.

155.

€

6 − 5 x( )2 156.

€

x 2 x + 1( ) x − 1( )

Algebra II Name: Page 16 of 16 157.

€

x + y( ) x 2 − 2 xy + y 2# $ % &

' ( 158. 3y − 2( ) y + 4( )

159.

€

x + y( )3 160.

€

4 x − 1( ) 1 + 4 x( ) Solve for n. 161.

€

2 n +1 = 2 6 162.

€

4 n + 3 = 16 n

163. Expand 2x − 3( )4

164. Expand a + 3b( )5

Divide.

165. 3x 3 − 5x 2 + x − 2

x − 2 166.

y 4 − 4y 2 + y + 4y + 2

Unit 4 – 4 thru 4 – 6 on Factoring is also on the midterm. Since you just had the test on that, I did not include it on this study guide.