Download Honors Algebra II Pre-Test

Name:___________________________
Pre-AP Geometry Summer Assignment
(Required Foundational Concepts)
Pre-AP Geometry is a rigorous critical thinking course. Our expectation is that each
student is fully prepared. Therefore, the following Algebra 1 concepts (part I and part II)
must be mastered prior to beginning Pre-AP Geometry.
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Solving linear equations
Graphing linear equations
Finding slope from ordered pairs and/or linear equations.
Writing equations of lines in slope-intercept, point-slope and standard forms
Solving systems of equations
Multiplying binomials
Factoring
Solving quadratic equations by factoring and with Quadratic Formula
Simplifying, multiplying, and adding radicals
Solving right triangles using the Pythagorean Theorem
Multiplying, dividing, adding and subtracting expressions with exponents, take
a power to a power, simplifying expressions with negative exponents
Adding, subtracting, multiplying, dividing, and simplifying fractions
Solving literal equations, etc…
Be prepared to turn in this assignment on the first day of school. You are expected to
show all your work in a clear and organized manner. For your benefit, the assignment
includes the answers. SGPHS staff will not be available for tutoring during the summer
so you may need to look up the concepts and related vocabulary through some of the
following helpful sites:
Helpful websites
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khanacademy.org
youtube.com (simply search the topic)
google.com
mathtv.com
A test over these concepts will take place during the first week of school. The test
will be non-calculator.
Leave answers in simplified radical form or improper fractions (no decimals).
I look forward to meeting you in August.
Mr. Roberson, Geometry PAP Teacher
PART I
This assignment should be completed without the use of a calculator. Show all work for credit.
Solve. Use improper fractions where appropriate. (No decimals or mixed numbers).
1. 4(3n + 5) – 2(2 – 4n) = 6 – 2n
3.
2
1
x 7
3
6
5. 2(4x) – (x – 1) = 2 (1 – x)
2. 3x – 12 – 5x = 5 – 6x – 9
4.
2 3
7 2
 x  x
15 5
15 3
6.
2
5 1
a  a4
3
6 2
2
Graph each line:
2
7. y =  x – 3
5
10
8. 3x – 2y = 12
y
10
x
-10
10
y
x
-10
10
-10
-10
9. y = 3
10. x = -1
5
y
5
y
x
-5
5
x
-5
-5
5
-5
Find the slope of each line:
11. y = -2x – 4
12. a horizontal line
13. a vertical line
14. y = -x
15. The line passing through A (-2, 3) and B (2,-4)
3
Write the equation of the line described.
16. Slope 2, y intercept –4
(Show answer in slope-intercept form.)
18. With undefined slope, passing through (2, 1)
17. Passing through the points (-1,3) and (5, 7)
(Show answer in standard form.)
3
19. Slope  , passing through the point (5, -2)
5
(Show answer in point-slope form.)
Solve each system of equations using addition (elimination) or substitution.
20. 2x – 3y = 8
21. 3y – 2x = 4
x + y= 4
1
(3y – 4x) = 1
6
22. 5x – 2y = 3
2x + 7y = 9
23. 2x – 3y = 1
3x + 5y = 11
4
Multiply.
24. (x – 3) (x + 7)
25. ( 2x – 1)(5x + 3)
26. (x + 8) 2
27. (2x – 3) 2
28. (x – 2) (x + 2)
29. (7m – 1)(2m – 3)
Factor.
30. a 2 + 9a + 18
31. 2a 2 + a – 15
32. 3y 2 – 14y – 24
33. b 2 – 8b + 16
34. x 2 – 81
35. 16p 2 – 25
Solve by factoring.
36. 3x 2 + 13x – 10 = 0
37. 2a 2 + 5a = -4(a + 1)
38. a2 – 4a = 21
5
Solve using the Quadratic Formula. Give exact answers in simplified radical form.
39. a 2 – 3a – 6 = 0
40. 2a 2 + 5a + 1 = 0
Simplify.
41. 45
42. 3 72
43. 5 32
44. 7 3  3 3
45. 3 6  24
46. 7 8  5 2
Use the Pythagorean Theorem to find the value of the variable. Give exact answers in simplified
radical form.
47.
48.
y
2
x
2
4
49. In little league baseball, the distance of the paths between each pair of consecutive bases is 60 feet
and the paths form right angles. How far does the ball need to travel if it is thrown from home
plate directly to second base?
6
Simplify. Use only positive exponents in your answers.
50. a5  a  a-2
16x 2 y
51.
2xy
52. (2n)4  (3n)2
53. (3x2y)2  (-4xy3)
54.
55. (
)
56. Find the area and perimeter of the rectangle.
A = __________________
(2a)2
P = __________________
Solve each literal equation for the stated variable.
57. Solve P = 2l + 2w for w
59. Solve V   r 2h for h
(3b2)3
58. Solve A =
1
bh for h
2
9
60. Solve F = C + 32 for C
5
7
Key:
Page 2
5
1. 11
2. 2
43
3.
4
4. –5
1
5.
9
6. – 19
Page 3
7.
1
y
-10
1
x
-10
y
1
8.
-
1
x
11. –2
12. 0
13. Undefined
14. –1
7
15. 4
Page 6
Page 4
16. y = 2x – 4
17. 2x – 3y = -11
18. x = 2
3
19. y  2    x  5 
5
20. (4, 0)
2
21. (-1, )
3
22. (1, 1)
23. (2, 1)
42. 18 2
-
5
x
y
5
10.
-
5
x
41. 3 5
43. 20 2
44. 4 3
45. 5 6
46. 140
47. 2 5
48. 14
49. 60 2 ft.
Page 7
52. 144n 6
25. 10x 2 + x – 3
53. -36x 5 y 5
26. x 2 + 16x + 64
54.
2
y
5  17
4
24. x 2 + 4x – 21
28. x 2 – 4
5
40.
Page 5
27. 4x – 12x + 9
9.
3  33
2
50. a 4
51. 8x
2
-
39.
29. 14m – 23m + 3
30. (a + 6)(a + 3)
31. (2a – 5)(a + 3)
32. (3y + 4)(y – 6)
33. (b – 4) 2
34. (x – 9)(x +9)
35. (4p – 5)(4p + 5)
2
36. x = or x = -5
3
1
37. a = -4 or a = 
2
38. a = 7 or a = -3
55.
56. A = 108a 2 b 6
P = 8a 2 + 54b 6
P  2l
57. w 
2
2A
58. h 
b
V
59. h  2
r
5
60. C  F  32 
9
-
8
PART II
(Part II is not required but is a great opportunity for additional practice!)
ORDER OF OPERATIONS
REMEMBER: PLEASE EXCUSE MY DEAR AUNT SALLY
P  Perform all operations that occur within grouping symbols such as ( ), { }, or [ ].
E  Evaluate exponents (powers and roots)
M & D  Perform multiplication and division operations from left to right
A & S  Perform addition and subtraction operations from left to right.
Simplify the following expressions.
1] (-4 + 2)(-2 + 5)2
2] 5 − 12 ÷ 3 − 7
3] 8 ÷ (6 − 2) + 5
4] 11 × 622 − 3
LINEAR EQUATIONS
Helpful hints:
• When solving a linear equation the goal is to isolate the variable.
• To move a term across the equal sign, you must use inverse operations.
•To keep the equation balanced, you must perform the operation on both sides of the equation.
Find the value of the variable.
1 ] 4 a = − 20
2] −
3] 2p + 5 = 13
4] 12 + 2b = 2 + 5b
6] 2(4x + 4) =x + 1
7] 2(x + 5) = 3 (x − 2)
8] 180 − x = 3 (90 −x)
1
(2x − 4)
2
10] 5x − [7 − (2x −1)] = 3 (x − 5) + 4(x + 3)
5
5] 4x + 5 + 5x +40 = 180
1
9]
1
2 (6 + 4x) − 4 (8x − 12) =
x
3
=5
FRACTIONS
Examples: Simplify the fraction.
a] 8w
8w
2
2
= 4w
Simplify the fraction.
1] 14
6]
11]
b]
5x - 10
5x - 10
15
15
2] 75
=
5 (x - 2)
15
=
x -2
3
-8y 3
10y 2
- 18r 3t
6a + 12
b 2 - 12b + 35
36 - x2
x
36
b 2 - 25
36 -x 2
x+6
4] 3 x
15
12 r t
x +6
3] 18 a
10
7]
c]
8]
12]
6
a 2 + 8a + 16
a 2 - 16
=
x+6
(6 - x)(6 + x)
=
1
(6 - x)
5]
5 bc
10b 2
x +2
9] 3x +6
10]
13]
5a + 5b
a 2-b 2
3x 2 - 6x - 24
3x 2 + 2x - 8
THE COORDINATE PLANE
Name the coordinates of each point.
1. M
2. N
3. K
4. R
5. S
6. T
7. U
8. V
9. W
10. Q
11. Name all the points shown that lie on the x–axis.
12. Name all the points shown on the y–axis.
13. What is the x–coordinate of every point that lies on a vertical line through P?
14. Which of the following points lie on a horizontal line through W?
(-2, 1)
(2, 3)
(1, -3)
(-2, 0)
(0, -3)
(2, 0)
Name all the points shown that lie in the quadrant indicated. (A point on an axis is not in any quadrant.)
15. Quadrant I
16. Quadrant II
17. Quadrant III
18. Quadrant IV
Plot each point on the graph shown to the right.
19. A (2, 1)
20. B (5, 0)
21. C (0, 3)
22. D (-3, 1)
23. E (-2, -1)
24. F (1, -2)
25. G (4, -2)
26. H (-4, -3)
Find the coordinates of the midpoint of 𝐴𝐵.
27. A (0, 1), B (4, 1)
28. A (-3, 4), B (-3, -4)
GRAPHS AND EQUATIONS OF LINES
SLOPE-INTERCEPT FORM
y = m x + b, where m = slope and b = y-intercept
Graphing Equations in Slope-Intercept Form
1. Write the equation in slope-intercept by solving for y
2. Find the y-intercept and use it to plot the point where the line crosses the y-axis.
3. Find the slope and use it to plot a second point on the line.
4. Draw a line through the two points.
WRITING EQUATIONS OF LINES
To write the equation of a line in slope-intercept form you must have the slope and the y-intercept.
Given a Graph
Identify the y-intercept and the slope from the graph and plug those values into the point-slope
equation y - y1 = m(x - x1).
Given the Slope and a Point
Example:
Write an equation of the line that passes through (1, 3) and has a slope of -5.
Step 1: Substitute -5 for m, 1 for x1, and 3 for y1.
Step 2: Distribute -5(x - 1).
Step 3: Solve for y by adding 3 to each side.
y - 3 = -5(x - 1)
y - 3 = -5x + 5
y = -5x + 8
Given Two Points
𝑦2−𝑦1
Step 1: Find the slope of the line using the two points and the formula m = 𝑥 −𝑥 .
2 1
Step 2: Choose either point and follow the steps above.
PROBLEMS:
Write an equation of the line that passes through the given point and has the given slope.
1. (0, 4), m = 2
2. (1, 0), m = 3
3. (9, 3), m = -23
Write an equation of the line that passes through the given points.
4. (8, 5), (11, 14)
5. (-5, 9), (4, 7)
Write an equation of the line.
7.
8.
6. (-8, 8), (0, 1)
SYSTEMS OF LINEAR EQUATIONS
REVIEW: SUBSTITUTION
METHOD
Solve: Y = 5 – 2X
5X – 6Y = 21
Solution: Substitute 5 – 2x for y.
5X – 6(5 - 2X) = 21
5X – 30 + 12X = 21
17X – 30 = 21
17X = 51
X =3
Solve each system using substitution.
1. y = 2x + 5
2. 8x + 3y = 26
3x - y = 4
2x = y - 4
3. x - 7y = 13
3x – 5y = 23
REVIEW:
ELIMINATION METHOD Example 1 ~
Solve: 3X + 4Y = -10
5X – 2Y = 18
Solution: 3X + 4Y = -10
2(5X – 2Y = 18)
Example 2 ~ 5x – 2y = -19
2x + 3y = 0
Solution: 3(5x – 2y = -19)
2(2x + 3y = 0)
Solve each system using elimination.
1. 3x + 4y = 9
2. 5x + 3y = 30
-3x – 2y = -3
3x + 3y = 18
5. 2x – 8y = 24
3x + 5y = 2
6. 5x – 9y = 47
6x + 2y = 18
Then substitute 3 for x. y = 5 – 2(3)
Y =-1
ANSWER: (3, -1)
4. 3x + y = 19
2x – 5y = -10
3x + 4y =
-10 10X – 4Y
13x= 36 = 26
x = 2 Then substitute 2 for x.
3(2) + 4y = -10
4Y =
-16 Y =
ANSWER: (2, -4)
15x -6y = -57
4x + 6y = 0
19x
= -57
x = -3
Then substitute -3 for x.
2(-3) + 3y = 0
-6 + 3y = 0
3y = 6
y =2
ANSWER: (-3, 2)
3. 3x + y = -3
x + 4y = 10
4. 4x – 6y = -26
-2x + 3y = 13
RADICAL EXPRESSIONS
Simplify.
1. √56
2.
Solution: √56 = √4 ∙ 14 = 2√14
3. �3√7�2
√7
√3
Solution:
Solution: (3√7)2 = �3√7 ∙ 3√7� = 3(3)�√7 ∙ √7� = 9(7) = 63
7
�3 =
√7
√3
∙
√3
√3
=
√21
√9
=
√21
3
Simplify the following.
1. √36
7.
80
25
13.
(2 3 )
2
2.
81
3.
24
8.
5
3
9.
2 13
12
15.
(9 2 )
14.
Solve for x.
2
2
2
1. 2 + x = 4
Solution: 4 + x
2
= 16
x =12
x = √12
x = 2√3
2
2
(3 8 )
4.
10.
2
98
5.
300
6.
205
48
11.
132
12.
16. 5 18
17. 4 27
1
4
(
2
Solve for x. Assume x represents a positive number.
1. 32 + 42 = x 2
2. x 2 + 42 = 52
3. 52 + x 2= 132
4. x 2 + 32 = 42
5. 42 + 72 = x 2
6. x 2 + 52 = 102
7. 12 + x 2 = 32
8. x 2 + 52 = 5 2
2
)
(
9. x 2 + 7 3
)
2
18. 6 24
2. x 2 + �3√2� = 92
Solution: x 2 + (9)(2) = 81
x 2 + 18 = 81
x 2 = 63
x = √63
x = √9 ∙ 7
x = 3√7
(
17
2
)
= (2x )2
EXPONENTS
REVIEW: Exponent Rules
a0=1
𝑎𝑚
=
𝑎𝑛
Example: 50 = 1
am – n Example:
𝑏7
𝑏3
7–3
=b
=b
am • an = am + n
4
a-m =
Example: x
1
𝑎𝑚
Example: 6 -2 =
1
62
1. (-6)3
2. (-5)4
3. 3-2
4. 2-3
8. 150
9. (-1)20
10. (-1)99
11. 23 • 22 • 2-4
Simplify. Use only positive exponents in your answers.
8
14. x
19. (b 4)2
-1
•x
-2
20. (s 5)3
•x
(am)n = a m(n) Example: (y 3)4 = y
=
15.
r
r
9
4
2
6.  
3
5. (-4)-3
16.
21. (3y 2)(2y 4)
=x
3(4)
−2
5
17. a • a
22. (4x 3y 2)(2y 4)
-1
4. 3X 2 + 54X + 243
(x – 14)(x + 2)
5. –X 2 + 2X – 1
12
−3
23. (5a 2b 3)(a -2b)
2. x2 – 12x - 28
Factor the trinomial. if the trinomial cannot be factored, say so.
1. X 2 + 5X + 4
2. X 2 – 8X + 12
6
18. (x 2)-2
FACTORING
the two terms.
=y
=x
5
7.  
3
24. (-2ab 5)(-4ab -3)
Examples: Factor.
1. 24X 3 – 32X 2
Hint: 8x 2 is the greatest
2
= 8X (3X – 4) common factor between
2+4
12. 42 • 33 • 2-3
m3
m
4
1
a6
Simplify.
13. r 5• r
2
3. 5X 2 + 5X -10
Example:
3x
2
+ 14x + 8 = 0
QUADRATIC EQUATIONS
Solution: (3x + 2)(x + 4) = 0
3x + 2 = 0 or x + 4 = 0
Solve each equation by factoring. (Hint: You may need the Quadratic Formula, from Algebra I. If you are not familiar
with the Quadratic Formula, please do some research on the web. If this doesn’t help, please email me and I will help.)
1. x
2
+ 5x - 6 = 0
2. x
4. x
2
+ 8x = 20
5. 4x2 + 15 = 17x
Example:
3 y
=
2 22
3(22) = 2y
66 = 2y
33 = y
1.
2.
2
– 7x – 18 = 0
10
6
=
6x + 7 2x + 9
5.
7.
2 − 4x 6 x − 8
=
−6
10
8.
2
= 20x – 36
6. 3x 2- 13x – 10 = 0
PROPORTIONS
x + 4 x −2
=
5
3
3(x + 4) = 5(x – 2)
3x + 12 = 5x – 10
22 = 2x
11 = x
Solve the following proportions using the format used in the examples.
7 y
7 21
2.
1.
=
=
2 3
3 x
4.
3. x
4
x −3
x +2
5
=
=
6
x +3
4
x +1
3.
25 10
=
15 x
6.
3x − 5 x − 15
=
2
4
9.
2
x −2
=
6
x −3
Helps
Order of Operations/PEMDAS
- simplify expressions
Solving Equations
Finding Slope and Equations of Lines
Solving Systems of Equations
Simplifying Radicals