Pre-AP Geometry Summer Assignment (Required Foundational Concepts)

Transcription

1 Name: Pre-AP Geometry Summer Assignment (Required Foundational Concepts) Pre-AP Geometry is a rigorous critical thinking course. Our epectation 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 epressions with eponents, take a power to a power, simplifying epressions with negative eponents 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 epected 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

2 PART I

3 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 mied numbers). 1. 4(3n + ) ( 4n) = 6 n. 3 1 = (4) ( 1) = (1 ) 6. 1 a a 4 3 6

4 Graph each line: 7. y = 3 10 y 8. 3 y = 1 10 y y = = -1 y -10 y Find the slope of each line: 11. y = a horizontal line 13. a vertical line 14. y = - 1. The line passing through A (-, 3) and B (,-4) 3

5 Write the equation of the line described. 16. Slope, y intercept 4 (Show answer in slope-intercept form.) 17. Passing through the points (-1,3) and (, 7) (Show answer in standard form.) 18. With undefined slope, passing through (, 1) 19. Slope 3, passing through the point (, -) (Show answer in point-slope form.) Solve each system of equations using addition (elimination) or substitution. 0. 3y = y = 4 + y = 4 1 (3y 4) = 1 6. y = 3 + 7y = y = y = 11 4

6 Multiply. 4. ( 3) ( + 7). ( 1)( + 3) 6. ( + 8) 7. ( 3) 8. ( ) ( + ) 9. (7m 1)(m 3) Factor. 30. a + 9a a + a y 14y b 8b p Solve by factoring = a + a = -4(a + 1) 38. a 4a = 1

7 Solve using the Quadratic Formula. Give eact answers in simplified radical form. 39. a 3a 6 = a + a + 1 = 0 Simplify Use the Pythagorean Theorem to find the value of the variable. Give eact answers in simplified radical form y 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

8 Simplify. Use only positive eponents in your answers. 0. a a a y y. (n) 4 (3n) 3. (3 y) (-4y 3 ) 4.. ( ) 6. Find the area and perimeter of the rectangle. A = (a) P = (3b ) 3 Solve each literal equation for the stated variable. 7. Solve P = l + w for w 8. Solve A = 1 bh for h 9. Solve V r h for h 60. Solve F = 9 C + 3 for C 7

9 Key: Page Page y y y y Undefined Page y = y = = 3 y 0. (4, 0) (-1, 3 ). (1, 1) 3. (, 1) Page m 3m (a + 6)(a + 3) 31. (a )(a + 3) 3. (3y + 4)(y 6) 33. (b 4) 34. ( 9)( +9) 3. (4p )(4p + ) 36. = 3 or = - Page ft. Page 7 0. a n y A = 108a b 6 P = 8a + 4b 6 P l 7. w 8. h A b 9. V h r C F a = -4 or a = a = 7 or a = -3-8

10 PART II (Part II is not required but is a great opportunity for additional practice!)

11 ORDER OF OPERATIONS REMEMBER: PLEASE EXCUSE MY DEAR AUNT SALLY P Perform all operations that occur within grouping symbols such as ( ), { }, or [ ]. E Evaluate eponents (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 epressions. 1] (-4 + )(- + ) ] ] 8 (6 ) + 4] 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 = 0 ] = 3] p + = 13 4] 1 + b = + b 3 ] = 180 6] (4 + 4) = + 1 7] ( + ) = 3 ( ) 8] 180 = 3 (90 ) 9] 1 1 (6 + 4) (8 1) = 1 ( 4) 4 10] [7 ( 1)] = 3 ( ) + 4( + 3) FRACTIONS Eamples: Simplify the fraction. a] 8w 8w = 4w ( - ) - b] = = c] = (6 - )(6 + ) 1 = (6 - ) Simplify the fraction. 1] 14 ] 7 3] 18 a 4] 3 ] bc 10b 6] - 8y 3 10y - 18r 3 + t 6a + 1 7] 8] 9] 10] 1 r t a + b a - b 11] b - a 1] + 8a ] b - 1b + 3 a

12 THE COORDINATE PLANE Name the coordinates of each point. 1. M. N 3. K 4. R. S 6. T 7. U 8. V 9. W 10. Q 11. Name all the points shown that lie on the ais. 1. Name all the points shown on the y ais. 13. What is the 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? (-, 1) (, 3) (1, -3) (-, 0) (0, -3) (, 0) Name all the points shown that lie in the quadrant indicated. (A point on an ais is not in any quadrant.) 1. Quadrant I 16. Quadrant II 17. Quadrant III 18. Quadrant IV Plot each point on the graph shown to the right. 19. A (, 1) 0. B (, 0) 1. C (0, 3). D (-3, 1) 3. E (-, -1) 4. F (1, -). G (4, -) 6. H (-4, -3) Find the coordinates of the midpoint of AB. 7. A (0, 1), B (4, 1) 8. A (-3, 4), B (-3, -4)

13 GRAPHS AND EQUATIONS OF LINES SLOPE-INTERCEPT FORM y = m + 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. Find the y-intercept and use it to plot the point where the line crosses the y-ais. 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 - y 1 = m( - 1 ). Given the Slope and a Point Eample: Write an equation of the line that passes through (1, 3) and has a slope of -. Step 1: Substitute - for m, 1 for 1, and 3 for y 1. Step : Distribute -( - 1). Step 3: Solve for y by adding 3 to each side. y - 3 = -( - 1) y - 3 = - + y = Given Two Points Step 1: Find the slope of the line using the two points and the formula m = y y 1 1. Step : 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 =. (1, 0), m = 3 3. (9, 3), m = -3 Write an equation of the line that passes through the given points. 4. (8, ), (11, 14). (-, 9), (4, 7) 6. (-8, 8), (0, 1) Write an equation of the line

14 REVIEW: SUBSTITUTION METHOD SYSTEMS OF LINEAR EQUATIONS Solve: Y = X Solution: Substitute for y. X 6Y = 1 X 6( - X) = 1 X X = 1 17X 30 = 1 17X = 1 X = 3 Then substitute 3 for. y = (3) Y = - 1 ANSWER: (3, -1) Solve each system using substitution. 1. y = y = y = y = y = 4 = y y = 3 y = -10 REVIEW: ELIMINATION METHOD Eample 1 ~ Solve: 3X + 4Y = -10 X Y = 18 Solution: 3X + 4Y = -10 (X Y = 18) Eample ~ y = -19 Solution: 3( y = -19) + 3y = 0 ( + 3y = 0) 3 + 4y = X 4Y 13 = 36 = 6 = Then substitute for. 3() + 4y = -10 4Y = -16 Y = ANSWER: (, -4) 1-6y = y = 0 19 = -7 = -3 Then substitute -3 for. (-3) + 3y = y = 0 3y = 6 y = ANSWER: (-3, ) Solve each system using elimination y = y = y = y = -6-3 y = y = y = y = 13. 8y = y = y = 6 + y = 18

15 RADICAL EXPRESSIONS Simplify Solution: 6 = 4 14 = Solution: 7 = 7 3 = 1 = Solution: (3 7) = = 3(3) 7 7 = 9(7) = 63 Simplify the following ( 17 ) 13. ( ) ( ) ( ) Solve for = 4. Solution: 4 + = = 9 Solution: + (9)() = 81 = = 81 = 1 = 63 = 3 = 63 = 9 7 = 3 7 Solve for. Assume represents a positive number =. + 4 = 3. + = = = 6. + = = = ( ) 9. + ( ) 7 3 = ( )

16 EXPONENTS REVIEW: Eponent Rules a 0 = 1 Eample: 0 = 1 a m a n = a m + n Eample: 4 = + 4 = 6 a m a n = a m n Eample: b7 3 = b 7 3 = b 4 (a m ) n = a m(n) Eample: (y 3 ) 4 = y 3(4) = y 1 b a -m = 1 a m Eample: 6 - = 1 6 = 1 a6 Simplify. 1. (-6) 3. (-) (-4) (-1) (-1) Simplify. Use only positive eponents in your answers. 13. r r r 4 r 16. m m a a ( ) (b 4 ) 0. (s ) 3 1. (3y )(y 4 ). (4 3 y )(y 4 ) 3. (a b 3 )(a - b) 4. (-ab )(-4ab -3 ) FACTORING Eamples: Factor. 1. 4X 3 3X Hint: 8 is the greatest. 1-8 = 8X (3X 4) common factor between ( 14)( + ) the two terms. Factor the trinomial. if the trinomial cannot be factored, say so. 1. X + X + 4. X 8X X + X X + 4X X + X 1

17 QUADRATIC EQUATIONS Eample: = 0 Solution: (3 + )( + 4) = = 0 or + 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 me and I will help.) = = 0 3. = = = = 0 PROPORTIONS Eample: 1. 3 y + 4 =. = 3 3() = y 3( + 4) = ( ) 66 = y = = y = 11 = Solve the following proportions using the format used in the eamples. 7 y =. = = = = = = = = 3 6

18 Helps Order of Operations/PEMDAS - simplify epressions Solving Equations Finding Slope and Equations of Lines Solving Systems of Equations Simplifying Radicals