Summer Math Requirement – Algebra II Review For students entering Pre-‐Calculus Theory or Pre-‐Calculus Honors
The purpose of this packet is to ensure that students are prepared for the quick pace of Pre-‐Calculus. The Topics contained in this packet are the core Algebra I and Algebra II concepts that students must understand to be successful in Pre-‐Calculus. There are 11 concepts addressed in this packet:
A. Distance & Midpoint B. Circles C. Linear Functions D. Graphing Functions E. Factoring F. Evaluating Functions G. Quadratic Formula H. Solving Inequalities I. Domain & Range J. Properties of Exponents K. Square Roots and Rationalizing Denominators For each concept listed, there is a problem set that needs to be completed. There are websites given below where you can go to find examples, videos, and extra problems that deal with that particular topic if you need assistance on working the problem sets. If you try a site and it is not working or no longer available, it is your responsibility to do a search for another site. It is suggested that you complete this practice set in the few weeks prior to returning to school so that the information will be fresh. Please bring this worksheet as well as all of your work/solutions ON ANOTHER SHEET OF PAPER to class with you on the FIRST DAY OF SCHOOL. You must show your work – just a list of answers will not be accepted. We will go over the material briefly and have a summative assessment (test) within the first few weeks of school on these topics. If it has been a few years since you have had Algebra II or if you are uncomfortable with any of the topics after completing the problem set, it is recommended that you find some additional practice from the websites below or elsewhere. http://www.purplemath.com/ http://www.wtamu.edu/academic/anns/mps/math/mathlab/ http://www.basic-‐mathematics.com/ https://www.khanacademy.org/math/trigonometry http://www.khanacademy.org/math/algebra2 http://www.regentsprep.org/Regents/math/ALGEBRA/math-‐ALGEBRA.htm http://patrickjmt.com/ http://www.mathsisfun.com/algebra/index-‐2.html Please email Julie Vandiver ([email protected]), Kristy Eason ([email protected]), Tiffany Roach ([email protected]), Emmaline Lewis ([email protected]) , or Brian Lim ([email protected]) f you have any questions. We look forward to a great year in PreCalculus!!
Please complete the following problems on ANOTHER SHEET OF PAPER. You should include the directions, problem, all work necessary and box in your final answer. These problems should be done within two weeks of the start of school (please do not do them at the beginning of summer). Bring all work with you to the first day of class.
Feel free to use a calculator to check your work but please understand that the expectation is that you can complete the following without the use of a calculator. A. Distance and Midpoint
Distance Formula = 2 22 1 2 1( ) ( )x x y y− + − Midpoint Formula = 1 2 1 2,
2 2x x y y+ +⎛ ⎞
⎜ ⎟⎝ ⎠
Problems:
Find the distance and midpoint between these points: 1. ( ) ( )2, 3 & 4,7− 2. ( ) ( )1, 10 & 3, 9− − − B. Circles
Formula of a Circle: 2 2 2( ) ( )x h y k r− + − = where (h, k) is the center and r is the radius.
Problems:
Find the standard form of the equation of the specified circle: 1. Center (-‐3, 9) Radius = 2 2. Endpoints of the diameter (8, 2), (4, 8)
3. Find the center and radius. Then graph the circle: 2 2( 2) ( 1) 16x y+ + − = C. Linear Functions
Parallel lines have the Same Slope Perpendicular lines have Opposite Reciprocal Slopes
Slope = 2 1
2 1
y ymx x
−=−
Standard form: Ax By C+ =
Slope-‐Intercept Form: y mx b= + Point-‐Slope form: 1 1( ) ( )y y m x x− = − Zero slope = horizontal line (y=#) Undefined slope = vertical line (x=#) Problems: Determine the slope of the line passing through these points. 1) ( ) ( )4, 1 & 8,2− − 2) ( ) ( )3, 1 & 3,4−
Write an equation for the line described in in the given information.
3) (5, 4) m = 23
− 4) ( )2,4− m = 3− 5) passes through ( )6, 3− − and ( )2, 5− −
6) ( )2, 1− m = 0 7) ( )1,5− and perpendicular to 2 5x y− + =
D. Graphing Functions
Problems: Sketch a graph of the following functions: (Graph paper on back of packet if needed)
1. 2 4 8x y− = 2. 3 5y x= − + 3. ( ) ( )22 13
y x+ = −
4. 2y x= 5. y x= 6. 3y x= 7. 3y x= 8. y x= 9. 3y = − 10. ( )22 4y x= − − + 11. 1 3y x= + − 12. 2 3y x= −
E. Factoring
Problems: Factor the following completely: 1. 22 11 15x x− + 2. 25 180x − 3. 25 25x x− 4. 2 6 40x x+ − 5. 2 2 8x x− − 6. 2 4x − 7. 23x 75− 8. 2 9 20+ +x x 9. 2 16−x
10. 24x 4x 15− − 11. 22x 7x 4+ − 12. 2x 2x 35− −
13. 2 12 36m m− + 14. 22 18p − 15. 26 15x x+ −
16. 321 35−x x 17. 215 2y y− − 18. 26 10 4c c− −
19. 23 12s − 20. 218 9 1z z+ + 21. 215 16 4− +r r
22. 3 23 12 4w w w− − + 23. 3 24 12 3y y y+ − − 24. 3 22 12 18− +n n n
F. Evaluating Functions Problems: Evaluate the function at the values given in a, b, and c. Simplify where possible. You should have an answer for part a, b, and c for each question.
1. ( ) 5 2f x x= − a. ( )3f b. ( )5f − c. ( )2f r −
2. ( ) 2 3h x x x= + a. ( )2h − b. ( )7h c. ( )h x−
3. ( )2 2, 01 4, 02
x xg x
x x
⎧ − ≤⎪= ⎨+ >⎪
⎩
a. ( )0g b. ( )10g − c. ( )2g
G. Quadratic Formula Quadratic Formula: 2 42
b b acxa
− ± −=
Problems: Solve using the quadratic formula. Answers should be in simplest radical form – no decimals.
1. 22 6 1 0x x+ + = 2. 23 5 2x x− =
H. Solving Inequalities
Problems: Solve and graph on a number line. 1. 4( 3) 44x + ≤ 2. 7(3 7) 21 50x x− − + ≥ 3. 4 16 or 12 144x x< > 4. 6 2 4 12x− < − <
I. Domain and Range
Problems: 1. Find the domain and range of the relation below then determine if the relation is a function. {( 3,2),( 2,0),( 1, 1),(2,1),(2,3),(4,1)}− − − − Determine the domain and range of the functions below: 2. Domain: __________ Range:___________ 3. Domain: __________ Range:__________
J. Properties of Exponents
Suppose m and n are positive integers and a and b are real numbers. Then the following properties hold.
Problems: Simplify the following using properties of exponents. Make sure all answers contain only positive exponents.
1. 3 24 2x x− ⋅ 2. 4 34 2n n−⋅ 3. 3 2 4 34 3a b a b− −⋅ 4. ( )234a
5. 3 24x xy⋅ 6. ( )404r 7. ( )443x 8. ( ) 24 32x y−−
9. 2
32yy
10. 4 4 3
2 3 4
23x y zx y z
− −
− 11. ( )43 4
32x xx
−
− 12. ( )2 22 2y y⋅
13. ( )
4
34
2
2
m
m
−
− 14.
43x
− 15. ( )
13 215xy 16. ( ) 253x
−−
K. Square Roots and Rationalizing Denominators
Problems: Simplify.
1. 4 3288x y 2. 1259
3. 53 128x− 4. 7 33
23
2502x yx y
Rationalize.
5. 82x
6. 5 35
7. 72 3
8. 12
= ≠0 1, 0a a +=gm n m na a a −=m
m nn
a aa
− = 1nna
a
( ) =n n nab a b ⎛ ⎞ =⎜ ⎟⎝ ⎠
n n
n
a ab b
( ) =nm mna a
1
nna a= ( )m m m
mn nn na a or a a= =