Algebra 2 – December Examination Mixed Review Problems
(−5) 2 − 4 ÷ 2 ⋅ 10
3 2 + 3 ⋅ 2(−1) 5
1.
Simplify.
2.
Determine the following product.
3.
4.
5.
a.
(5x + 1)(2x − 7)
c.
(x
m
b.
(3 − 2x) 2
− 5)(9x m + 4 )
Name the property illustrated by the following:
a.
2 ⋅ (6 ⋅ −4) = (2 ⋅ 6) ⋅ −4
_____________________________
b.
2 ⋅ (6 ⋅ −4) = (6 ⋅ −4) ⋅ 2
_____________________________
Factor completely.
a.
6x 3 + 21x 2 − 8x − 28
b.
16x 2 − 8x −15
c.
3x 3 + 81
d.
x 4 − 26x 2 + 25
b.
x −6 yz 0
− 4 y −5
Simplify and write with positive exponents.
a.
⎛ (3x 3 y 4 ) 4 ⎞
⎜⎜ 8 −3 ⎟⎟
⎝ x y
⎠
6.
Simplify.
7.
Simplify.
8.
Simplify.
9.
Simplify.
Page 1 of 9
−1
75 x 3 y 5
2 10(3 15 − 3 2)
2x
12
5 −3
5 +1
2−1 + 2
2−3
10.
Simplify:
11.
Sketch the graph of the line 3x – 5y = 6. Include the x- and y- intercepts. (Below is what you
would receive on the exam – just a blank set of axes)
12.
Find the equation of the line through (3, –1) and (5, 2) in
a. point-slope form
b. standard form
13.
Find an equation of the line perpendicular to the line 2x + 7y = 16 with x-intercept (6, 0).
Your equation may be in any form.
14.
Solve the formula A − Dt = t + 13 for t .
15.
Simplify.
a.
c.
16.
− 3 − 80
b.
(3 − 4i)(2 − 5i)
− 2 ⋅ −8
Kelly Clarkson prices her personal body scent perfume (KC SWEET SCENT) by raising the
wholesale price 50% and then adding $20. What is the wholesale price of a bottle of KC
SWEET SCENT that sells for $365?
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54 − 2 y 3
17.
Factor completely:
18.
Simplify and write with positive exponents:
19.
Simplify.
20.
Simplify:
21.
Find an equation of the line parallel to the line 2x – 7y = 1 that passes through the point (2, –5).
Your equation may be in any form.
22.
Solve by completing the square to obtain exact solutions:
23.
Find the zeros of the function. Give exact answers.
f ( x) = 2 x 2 − 4 x − 5
24.
Solve by graphing:
⎧2 x + 3 y = 6
⎨
⎩2 x − y = −10
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− 2a −3b 0 c 7
ac −2
2−3 3
4+ 3
2 − 12 + 4 − 3
x 2 − 10x + 7 = 0
25.
Solve, using any method:
⎧2 x − 5 y = 20
⎪
⎨
1
⎪⎩ y = − 2 x + 2
26.
Over break, Mr. Rasnick and Ms. Blenko canoed up the Niagara River. They went the same
distance but Ms. Blenko took 24 less hours for the trip. Mr. Rasnick canoed at 5mph while Ms.
Blenko canoed at 25mph. How far was their canoe trip?
27.
Match each statement with the appropriate property:
Statement
Property
a)
2(x + y) = 2x + 2y
i. associative property of multiplication
b)
3⋅ 13 = 1
ii. commutative property of multiplication
c)
3⋅1 = 3
iii. multiplicative identity property
d)
3⋅ (2 ⋅ 4) = (3⋅ 2) ⋅ 4
iv. distributive property
e)
3⋅ 2 = 2 ⋅ 3
v. multiplicative inverse property
28.
Simplify each. Write using only positive exponents.
a)
−4 2 a17
9a−8
b)
2(−3c 5 d −4 )
c)
⎛ −2x 5 y −4 ⎞−3
⎜
⎟
−2
⎝ x
⎠
d)
(b ) (b )
29.
Multiply
a)
(2x
n
− 3)( 3x n + 2)
b)
y 2
(3y − 2x )
2
x−2 y
2
c)
30.
Factor each completely:
a)
x 4 + 5x 2 − 36
b)
6x 3 − 2x 2 − 3x + 1
c)
12a 2b + 21ab − 45b
d)
8w 7 − 27wx 3
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( x − 2)(2x 2 − x + 3)
31.
Find the slope of a line containing the points (–10, 7) and ( –10, –7).
32.
Graph 2y – 3x = – 6. State the slope of this line.
33.
Your text messaging plan costs a flat $2.95 per month plus $0.05 per text message sent or
received. Write a function that models this situation for x messages sent and received. Then
determine how much you would pay in a month where you sent and received a total of 140
messages.
34.
A line passes through the points ( –5, 4) and (3, –2). Find the equation for this line in standard
form, slope-intercept form and point-slope form.
35.
Given the line 2x – 5y = 4, find the equations in standard form of the lines parallel and
perpendicular to this line through the point (2,– 2).
36.
Simplify.
a)
3
48 x 5 y 4 z 14
b)
24 x 4 y 7 ⋅ 21xy 3
c)
3
16x4 y7 ⋅ 3 20 x2 y
d)
8 + 2 20 − 18
e)
(2+
Page 5 of 9
10
)( 5 − 2)
f)
3
x2 ⋅ x
4−3 5
37.
Rationalize the denominator.
38.
Solve.
a)
6 − (3 x − 7 ) = 4(3 x − 5)− 11
c)
p(p + 10 ) = −21
39.
Find the zero(s) of the function:
3+ 2
b)
a) f(x)= –12x +16
40.
21 y − 14 y 2 = 0
b)
f(x)= 6x 2 −13x − 5
Simplify:
a)
(4 +
−81 − −6 − −144
) (
c)
(7 −
−16 2 + −9
)(
)
(−4 + i)(3 − 2i)
)
41.
Solve: 5 + x + 7 = x
42.
Solve for T: R =
43.
Solve by Completing the Square:
44.
Solve each by any algebraic method.
1
(T + RT + 300)
9
a.
⎧ 3x + 3y = 6
⎨
⎩5x − 6y = 15
c.
⎧ x + y = 4(y + 2)
⎨
⎩ x − y = 2(y + 4)
Page 6 of 9
b)
2 x 2 − 5x + 7 = 0
b.
⎧
1
⎪ y = x −1
⎨
2
⎪⎩ x = 4 + 2y
d.
⎧ x + 6y + 3z = 4
⎪
⎨2x + y + 2z = 3
⎪ 3x + z − 2y = 0
⎩
45.
[calc ok] You borrowed $30,000 to pay for your Hummer. Part of this money was borrowed at
8%, part at 10% and part at 12%. The annual interest cost was $3040, and the total amount
borrowed at 8% and 10% combined was twice the amount borrowed at 12%. How much was
borrowed at each rate?
46.
[calc ok] At Pop This!, cheese popcorn worth $2.50 pound is mixed with peppermint popcorn
worth $7.50 per pound in order to get a 20lb mixture worth $4.50 per pound. How much of each
popcorn is used?
47.
Randall Rents Rusty Recks, Inc. charges a fixed amount per weekly rental plus a charge for each
mile driven. A one-week trip of 520 miles costs $250, and a two-week trip of 800 miles costs
$440. Find the weekly charge and the charge for each mile driven.
48.
Graph y = x. Label at least two points.
49.
Find the slope of the line containing (9, –4) and (6, –5).
50.
Write the equation of the line in standard form parallel to y = 2 x + 9 and passing through (-5, 0)
51.
Solve:
⎧36 + 7 x − 8 y = 0
⎨
⎩−10 y = −12 − 6 x
52.
Simplify:
(−3x −1z 5 )−2
53.
Simplify:
54.
Simplify:
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(
30 6a −1b 2
a −6b 6
−2 2
)
−2
55.
Solve:
3x ( 5 − 3x ) = 4
56.
Graph:
1
y − 2 = ( x − 2)
3
57.
Sketch a graph of the quadratic, including the vertex and four other well chosen points.
f ( x) =
58.
1
2
( x − 3) − 2
2
Find the vertex of the following quadratic by completing the square AND by using the vertex
formula:
g ( x) = 2 x 2 − 16 x + 23
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59.
What is the equation for the axis of symmetry of f ( x) = 2 x 2 + 6 x + 8 ?
60.
If you are asked to write the equation for a line and you are given two points, which form of a
linear equation would be easiest to use?
61.
What is true about the graph of a system of equations that has no solution?
62.
Without a calculator, how would you determine if 3 17 − 11 will be a positive or negative
number?
63.
The Yo Mama Charter Company transports algebra students from Boston to Irrational Island
which is off the coast of Maine. Currently, they are charging $250 per passenger and carry 200
passengers per day. Taylor Swift, the manager of Yo Mama, estimates that the company will
lose 10 passengers for each increase of $25 in the fare.
a) Find an income function for x number of price increases.
b) If the income earned is $54,000, how many price increases have their been?
c) What price would you tell Yo Mama to charge in order to maximize their income?
64.
(calc ok) Mathea determined that the path a football travels when it is being kicked for a field
goal is modeled by the following function:
4
x2
where h(x) is the ball’s height in feet and x is the distance traveled.
h( x ) = x −
3
90
a) What is the maximum height the football will reach?
b) After what distance traveled does the ball return to the field?
65.
66.
A rectangle has a width 4 feet longer than its length.
a)
Write a function A(w) for the area of this rectangle.
b)
Does A(w) have a maximum, minimum or neither? Explain.
Find the zeros.
Page 9 of 9
f ( x) = −3 ( x + 2) + 1
2