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.

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

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

Name:___________________________

PART I

2

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

2. 3x – 12 – 5x = 5 – 6x – 9

3. 2 1

73 6

x

4. 2 3 7 2

15 5 15 3x x

5. 2(4x) – (x – 1) = 2 (1 – x)

6. 2 5 1

43 6 2

a a

3

Graph each line:

7. y = 2

5 x – 3

8. 3x – 2y = 12

9. y = 3 10. x = -1

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)

y

x 10

10

-10

-10

y

x 10

10

-10

-10

y

x 5

5

-5

-5

y

x 5

5

-5

-5

4

Write the equation of the line described.

16. Slope 2, y intercept –4 (Show answer in slope-intercept form.)

17. Passing through the points (-1,3) and (5, 7) (Show answer in standard form.)

18. With undefined slope, passing through (2, 1) 19. Slope

3

5, passing through the point (5, -2)

(Show answer in point-slope form.)

Solve each system of equations using addition (elimination) or substitution.

20. 2x – 3y = 8 x + y = 4

21. 3y – 2x = 4

1

6(3y – 4x) = 1

22. 5x – 2y = 3 2x + 7y = 9

23. 2x – 3y = 1 3x + 5y = 11

5

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

6

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.

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?

4

y 2

2 x

7

Simplify. Use only positive exponents in your answers.

50. a5 a a-2

51. 216x y

2xy

52. (2n)4 (3n)2

53. (3x2y)2 (-4xy3)

54.

55. (

)

56. Find the area and perimeter of the rectangle. A = __________________ P = __________________

Solve each literal equation for the stated variable.

57. Solve P = 2l + 2w for w 58. Solve A =

2

1bh for h

59. Solve 2V r h for h 60. Solve F =

5

9C + 32 for C

(2a)2

(3b2)3

8

Key: Page 2

1. -5

11

2. 2

3. 43

4

4. –5

5. 1

9

6. – 19

Page 3

7.

8.

9.

10.

11. –2

12. 0

13. Undefined

14. –1

15. -7

4

Page 4

16. y = 2x – 4

17. 2x – 3y = -11

18. x = 2

19. 3

y 2 x 55

20. (4, 0)

21. (-1, 2

3)

22. (1, 1)

23. (2, 1)

Page 5

24. x 2 + 4x – 21

25. 10x 2 + x – 3

26. x 2 + 16x + 64

27. 4x 2 – 12x + 9

28. x 2 – 4

29. 14m 2 – 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)

36. x = 2

3 or x = -5

37. a = -4 or a = 1

2

38. a = 7 or a = -3

Page 6

39. 3 33

2

40. 5 17

4

41. 3 5

42. 18 2

43. 20 2

44. 4 3

45. 5 6

46. 140

47. 2 5

48. 14

49. 60 2 ft.

Page 7

50. a 4

51. 8x

52. 144n 6

53. -36x 5 y 5

54.

55.

56. A = 108a 2 b 6

P = 8a 2 + 54b 6

57.

2

2

P lw

58. 2A

hb

59.

2

Vh

r

60. 5

C F 329

y

x 1

1

-10

-10

y

x 1

1

-

-

y

x 5

5

-

-

y

x 5

5

-

-

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] − = 5 3] 2p + 5 = 13 4] 12 + 2b = 2 + 5b

5] 4x + 5 + 5x +40 = 180 6] 2(4x + 4) = x + 1 7] 2(x + 5) = 3 (x − 2) 8] 180 − x = 3 (90 −x)

9] (6 + 4x) − (8x − 12) = 1 (2x − 4) 10] 5x − [7 − (2x −1)] = 3 (x − 5) + 4(x + 3)

FRACTIONS Examples: Simplify the fraction.

a] 8 w 8 w = 4 w b] c]

Simplify the fraction. 1] 2] 3] 18 a 4] 5]

6] - y 32

810y

7] 8] 9] 10]

11] 12] 13]

2

5x - 10

15

x + 6

36 - x 2

5

x

3

1

2

1

4 2

2

5x - 10

15 =

5 (x - 2)

15=

x - 2

3

x + 6

36 - x2 =

x + 6

(6 - x)(6 + x)=

1

(6 - x)

14

10

75

15 36 3 x

x

5 bc

10b 2

- 18r 3t

12 r t

6a + 12

6

x + 2 3x + 6

5a + 5b

a 2 - b 2

b 2 - 25

b 2 - 12b + 35

a 2 + 8a + 16

a 2 - 16

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 LINESSLOPE-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 y2. 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 + 5y = -5x + 8

Given Two Points

Step 1: Find the slope of the line using the two points and the formula m = 𝑦2−𝑦1𝑥2−𝑥1

. 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) 6. (-8, 8), (0, 1)

Write an equation of the line. 7. 8.

SYSTEMS OF LINEAR EQUATIONS REVIEW: SUBSTITUTIONMETHOD

Solve: Y = 5 – 2X Solution: Substitute 5 – 2x for y. 5X – 6Y = 21 5X – 6(5 - 2X) = 21

5X – 30 + 12X = 2117X – 30 = 21

17X = 51 X = 3 Then substitute 3 for x. y = 5 – 2(3)

Y = - 1 ANSWER: (3, -1)

Solve each system using substitution. 1. y = 2x + 5 2. 3. x - 7y = 13 4. 3x + y = 19

3x - y = 48x + 3y = 26 2x = y - 4 3x – 5y = 23 2x – 5y = -10

REVIEW: ELIMINATION METHOD Example 1 ~ Solve: 3X + 4Y = -10

5X – 2Y = 18 Solution: 3X + 4Y = -10

2(5X – 2Y = 18) 3x + 4y =

-10 10X – 4Y= 36 13x = 26 x = 2 Then substitute 2 for x.

3(2) + 4y = -10 4Y =

-16 Y =ANSWER: (2, -4)

Example 2 ~ 5x – 2y = -19 Solution: 3(5x – 2y = -19) 2x + 3y = 0 2(2x + 3y = 0)

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) Solve each system using elimination. 1. 3x + 4y = 9 2. 5x + 3y = 30 3. 3x + y = -3 4.

-3x – 2y = -3 3x + 3y = 18 x + 4y = 104x – 6y = -26 -2x + 3y = 13

5. 2x – 8y = 24 6. 5x – 9y = 473x + 5y = 2 6x + 2y = 18

RADICAL EXPRESSIONS Simplify.

1. √56 Solution: √56 = √4 ∙ 14 = 2√14 2. √7√3 Solution: �7

3= √7

√3∙ √3√3

= √21√9

= √213

3. �3√7�2 Solution: (3√7)2 = �3√7 ∙ 3√7� = 3(3)�√7 ∙ √7� = 9(7) = 63

Simplify the following.

1. √36 2. 81 3. 24 4. 98 5. 300 6. 14

7.8025 8. 9.

2 1312

10. 20548 11. 213 12. ( )217

13. ( )22 3 14. ( )23 8 15. ( )29 2 16. 5 18 17. 4 27 18. 6 24

Solve for x. 1. 2.22 + x 2 = 42

Solution: 4 + x 2 = 16 x 2 + �3√2�

2= 92

Solution: x 2 + (9)(2) = 81 x 2 =12 x 2 + 18 = 81 x = √12 x 2 = 63 x = 2√3 x = √63

x = √9 ∙ 7 x = 3√7

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 = ( )25 2 9. x 2 + ( )27 3 = (2x )2

53

EXPONENTS REVIEW: Exponent Rules

a 0 = 1 Example: 50 = 1 am • an = am + n Example: x 2 • x 4 = x 2 + 4 = x 6

𝑎𝑚

𝑎𝑛 𝑏 = am – n Example: 𝑏7

3 = b 7 – 3 = b 4 (am)n = a m(n) Example: (y 3)4 = y 3(4) = y 12

a-m = 1𝑎𝑚 6 Example: 6 -2 = 1

2 = 1a6

Simplify.

1. (-6)3 2. (-5)4 3. 3-2 4. 2-3 5. (-4)-3 6.−

223

7. −

353

8. 150 9. (-1)20 10. (-1)99 11. 23 • 22 • 2-4 12. 42 • 33 • 2-3

Simplify. Use only positive exponents in your answers.

13. r 5• r 8 14. x -1 • x -2 15. 9

4rr

16. 3

5mm

17. a • a -1 18. (x 2)-2

19. (b 4)2 20. (s 5)3 21. (3y 2)(2y 4) 22. (4x 3y 2)(2y 4) 23. (5a 2b 3)(a -2b)

24. (-2ab 5)(-4ab -3)

FACTORING Examples: Factor. 1. 24X 3 – 32X 2 Hint: 8x 2 is the greatest 2. x2 – 12x - 28

= 8X 2(3X – 4) common factor between (x – 14)(x + 2) the two terms.

Factor the trinomial. if the trinomial cannot be factored, say so. 1. X 2 + 5X + 4 2. X 2 – 8X + 12 3. 5X 2 + 5X -10

4. 3X 2 + 54X + 243 5. –X 2 + 2X – 1

QUADRATIC EQUATIONS Example: 3x 2 + 14x + 8 = 0 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 2 – 7x – 18 = 0 3. x 2 = 20x – 36

4. x 2 + 8x = 20 5. 4x2 + 15 = 17x 6. 3x 2- 13x – 10 = 0

PROPORTIONS

Example: 1. =32 22

y 2. + −=4 2

5 3x x

3(22) = 2y 3(x + 4) = 5(x – 2) 66 = 2y 3x + 12 = 5x – 10

33 = y 22 = 2x 11 = x

Solve the following proportions using the format used in the examples.

1. =72 3

y2. =7 21

3 x3. =25 10

15 x

4. =+ +

10 66 7 2 9x x

5. =− +4 6

3 3x x6. − −

=3 5 15

2 4x x

7. − −=

−2 4 6 8

6 10x x 8. +

=+

2 45 1

xx

9. −=

−2 2

3 6x

x

HelpsOrder of Operations/PEMDAS

- simplify expressions

Solving Equations

Finding Slope and Equations of Lines

Solving Systems of Equations

Simplifying Radicals