AICE Mathematics AS Level Summer Review

Transcription

1 AICE Mathematics AS Level Summer Review Welcome to AICE mathematics. This is a rigorous and fast paced course so it is important to make sure that you have mastered the prerequisite skills. These necessary skills have been taught in previous mathematics courses and it is vital that you understand these in order to be successful in this course. We will not be reviewing these at the beginning of the course and so I have provided this packet as a review so that you can make sure that you fully understand these topics. It is imperative that you take the time to work through these problems, check your answers, and study anything you may have forgotten, or any skill that needs more practice. You will have a quiz on this material the first week of school and the assignment from this packet will be collected. I AM HERE TO HELP!! I look forward to having you in class net year and I want you to be successful. I have provided information on where to find review videos (made by me) on each topic as well as selected answers to check your work. If you need further assistance or have any questions regarding the course or this packet, please don t hesitate to contact me. You may do this through remind.com (see instructions below) or by at stacy.erb@polk-fl.net SIGN UP FOR REMIND MESSAGES Remind is a service that allows me to send reminders to you or you to contact me while our phone numbers remain anonymous. I often use this service to send announcements and reminders to my classes. It would be very beneficial for you to go ahead and sign up in case I send anything out over the summer. Feel free to contact me through this service over the summer if you have any questions. To sign up:

2 Calculator Information For the eam at the end of this course, you will be allowed to use a scientific calculator. Therefore, we will use one each day in this class so that you are very familiar with all of the functions on your calculator and can use it efficiently. You may use one on this material as well. However, you will still need to show each step of the process to receive credit. You will need to have your own scientific calculator with you in class each day. I highly recommend getting the TI-36Xpro because it is one of the easiest to use with the most functionality. TI-3XS is another good one. Almost any scientific calculator will be allowed on the eam but many do not have the same functions as those listed above. The TI-36Xpro is pictured below and can be picked up at Walmart, Target, Amazon and most other major stores. Additional Supplies 3 ring binder (at least 1.5 inch) 5-subject spiral notebook (college ruled) Notebook paper (college ruled) Blue/black ink pens Pencils

3 Video Tutorials for Review You can find links to video tutorials for review on the class website: You must have a free account to access the website. If you have not already done so, you can sign up using the information below: You may also access the videos on my You Tube channel at Erb Math.

4 MUST KNOW!! (formulas must be memorized) Slope (which will be called gradient in this course) y y m 1 1 E: find the slope of the line through the points (1, -5) and (-3, 7) Midpoint 7 ( 5) 1 m , y y 1 1 E: find the midpoint of the segment with endpoints (1, -5) and (-3, 7) 1 ( 3) ( 5) 7, Distance (also used to find length) d ( ) ( y y ) 1 1 E: Find the distance between (1, -5) and (-3, 7) d (( 3) 1) (7 ( 5)) ( 4) (1) Parallel Lines Have EXACTLY the same gradient (slope) E: if line A has a gradient of and is 3 parallel to line, what is the gradient of line?, = (-1, 1) It must be 3 Perpendicular Lines Have opposite reciprocal gradients. (This means you must flip the fraction and change the sign to its opposite) If two lines are perpendicular, then the product of their gradients will be -1 m1 m 1 E: if line A has a gradient of B? 3 and is perpendicular to line B, what is the gradient of line 4 B must have a slope of 4 3. (fraction is flipped, sign is opposite) Check m1 m

5 Perpendicular Bisector A perpendicular bisector both bisects a segment and is perpendicular to it Bisect (goes through the midpoint) Perpendicular (has opposite reciprocal slopes) A l 1 l 1 Line is a perpendicular bisector through segment AB B Equations of Lines Point-slope form (this is the form we will use most) y y m( ) which passes through the point (1, y1) and has gradient m 1 1 Horizontal lines (slopes are ) Eamples: y =, y = 3 Vertical lines (slopes are undefined) Eamples: = 4, = 7 You should be able to take point-slope form from above and write it in either of the two forms below. Slope-Intercept form ym b, where m is gradient and b is the y-intercept Standard form aby c, where a, b, and c are non-fraction constants and a is positive Intercepts intercepts o the coordinate where the function crosses the ais o to find the intercept, plug into y and solve for o written as (, -value) e: (, 4) y intercepts o the coordinate where the function crosses the y ais o to find the y intercept, plug into and solve for y o written as (, y-value) e: (5, )

6 Systems of Equations This is a very important topic that you will use throughout the entire course. Points of intersection to find points of intersection between two lines, or between a line and a curve, get one variable in one equation by itself and then substitute (plug in) this value into the other equation E: Find the coordinates of the point of intersection of the lines y = 4 and 3 + y = -1. You should notice that solving for y in the first equation is easiest. When you do this you get y = 4. Take this value, 4, and plug it into y in the second equation. You would then have 3 + ( 4) = -1. Solve for. You should find that = 1 and y = -. So the coordinates of the point of intersection is (1, -) Factoring and Completing the Square are also very important topics. For eamples, see tutorial videos.

7 Assignment: Complete the following problems on a separate sheet of paper. All work must be shown. Circle each answer. For questions 1 6, complete parts a c for the line described: a) write the equation of the line in point-slope form b) write the equation in slope-intercept form c) write the equation in standard form 1) Passes through the points ( -1, -1) and (4, -3) ) Passes through the points (-3, -3) and (, -5) 3) Passes through (5, ) and is parallel to the line 6 y 5 4) Passes through (-5, -4) and is perpendicular to y 4 3 5) Passes through (-1, 4) and is perpendicular to the line + 3y = 8. 6) Passes through (3, 9) and is parallel to the line joining (3, -1) and (-5, ). 7) Line A passes through the points (-6, 14) and (3, 1). a) Find the gradient of line A. b) Write the equation of line A in point-slope form. c) What is the relationship (parallel or perpendicular) of line A to the line with equation y 9 3 7?

8 8) Line A passes through the points (-4, -) and (6, -7). a) Find the gradient of line A. b) Write the equation of line A in point-slope form. c) Write the equation of line A in slope-intercept form. d) Write the equation of a line that is parallel to line A. e) Find the distance between the two points. f) Find the midpoint of the two points. 9) Line C is a perpendicular bisector of segment AB. The gradient of segment AB is -3 and point A has coordinates (4, 7). The point (, -1) lies on line C. a) What is the gradient of line C? b) Write the equation of line C. c) Write the equation of the line through points A and B. d) Find the coordinates of the point where line C intersects segment AB. e) Find the length of segment AB. This problem may be challenging! Drawing a picture and considering what you are given, and what you need, might help. 1) Determine if the triangle formed by the points (-3, 4), (1, 7), and (6, 8) is a right triangle. Justify your answer. 11) Find the equation of the line that passes through the points (5, -) and (-4, 4), giving your answer in the form a + by + c =. Hence find the coordinates where the line intersects the - ais. (Remember that the coordinate where a line crosses the ais is called the intercept and it is found by plugging into y) 1) The coordinates of the line segment AB are (4, 5) and (6, -4). Find the equation of the perpendicular bisector of AB, giving your answer in the form a + by + c =.

9 13) Find the equation of the perpendicular bisector of the segment joining the points (-, 6) and (7, -1). Solve each system by substitution. 14) 6 y 8 4 y 7 15) 3 y 13 4 y 8 16) 4 y 1 7 4y 4 17) y 7 3y1 18) y31 y41 19) y3 4 y 6 ) Find the coordinates of the point where line + 3y = 7 intersects line 4 y = 1. Solve the following equations by factoring. 1) 1118 ) 3 5 3) ) ) ) ) 3) ) 3 31) ) 9 36 Write each of the following in completed square form. State the following: a) verte b) ais of symmetry c) range 3) 68 33) ) 15 35) 63 36) ) 4 38) ) ) ) ) ) 5 148

10 Selected Answers For answers in point slope form, the first point was used. It is ok if you used the second point however, those answers are not listed here. 1) a) 3) a) 5) a) y1 ( 1) 5 b) 6 y ( 5) 5 b) 3 y 4 ( 1) b) y c) 5y7 6 y 4 c) 65y 5 y 3 11 c) 3y11 7) a) 7 9 b) y 7 14 ( 6) 9 c) the lines are perpendicular because they have 9 7 opposite reciprocal slopes. Also, ) Equation of perpendicular bisector is 418y11 14) (-3, -5) 16) (, 6) 1) ( 9)( ), solutions are 9 and 3) (3 ), solution is /3 5) 3(3 +5)( ), solutions are -5/3 and 7) -( 1)(3 1), solutions are, 1, and 3/1 9) -9( )( + ), solutions are, and - 31) 16 ( + )( ), solutions are, -, and 35) ( 1) + 64, verte: (1, 64), ais of symmetry: = 1, range: y 64 37) -( + 5) + 8, verte: (-5, 8), ais of symmetry: = -5, range: y ) 5( ), verte: 3 64 (, ) 5 5 of symmetry: = 3 5, range: y ) 6( ), verte: ais of symmetry: 3 4, ais 3 43 (, ), , range y 33) ( + 7) 11, verte: (-7, -11), ais of symmetry: = -7, range: y -11

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