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Lyman Memorial High School
Honors Pre-Calculus Prerequisite Packet
Fall 2017
Name:__________________________
Dear Honors Pre-Calculus Student, Within this packet you will find mathematical concepts and skills covered in Algebra I, II and Geometry. These concepts need to be reviewed and practiced throughout the summer. The completion of this review packet is very important and essential for your success in Pre-Calculus. These skills are used frequently throughout this course. Honors Pre-Calculus is a rigorous and fast-paced course. There will be extensive use of graphing calculators which is required for this course. A TI-84 Plus graphing calculator is recommended. Any other type of graphing calculator will have to be approved by the teacher. For this prerequisite packet, calculators should be used only to check work. The Pre-Calculus prerequisite packet is due the first day of school. It will be graded and it will count as a test grade. Work must be shown to support all answers. Your test grade will reflect both, your effort (50%) which is based on attempting all problems and showing work for all problems, and accuracy (50%). The packet is broken into specific concepts. Some sections have worked out examples followed by problems for you to complete. Be sure to complete each numbered exercise included in this packet. Below are a few websites you may wish to visit for additional examples and support. Algebra1online: http://teachers.henrico.k12.va.us/math/hcpsalgebra1/modules.html Algebra 2 online: http://teachers.henrico.k12.va.us/math/hcpsalgebra2/modules.html Algebra Help: http://www.algebrahelp.com/ Geometry: http://www.khanacademy.org/ Results from the summer prerequisite work will help guide skill and concept reinforcement lessons that will take place the first few weeks of school. Have a nice summer,
Lyman Math Department
Part 1 – Lines and Coordinate Geometry
1) Find an equation of the line in slope-intercept form that passes through (2,1)and (1,6).
2) Write the equation of the line parallel to the line 4𝑥 − 6𝑦 = −1 in point-slope form passing through the point (-10,2).
3) Write the equation of the line in slope-intercept form passing through the point (2, −4) and perpendicular to the line 𝑥 − 2𝑦 = 7
4) Find the value of 𝑎 if a line containing the point (𝑎, −3𝑎) has a y-intercept of 7 and a
slope of −2
3.
5) Find the distance between the points and then find the midpoint of the segment that joins them.
a) (0,8) and (6,16)
b) (−2,5) and (10,0)
Algebra Concepts
Slope-intercept form of a line 𝑦 = 𝑚𝑥 + 𝑏
Standard form of a line 𝐴𝑥 + 𝐵𝑦 = 𝐶
Point-slope form of a line 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)
Slope of a line 𝑚 =𝑦2−𝑦1
𝑥2−𝑥1
Geometry Concepts
Midpoint formula (𝑥1+𝑥2
2,
𝑦1+𝑦2
2)
Distance formula 𝑑 = √(𝑥1 − 𝑥2)2 + (𝑦1 − 𝑦2)2
Perpendicular bisector – a perpendicular line passing
through the midpoint of a segment.
Altitude of a triangle – a segment from a vertex
perpendicular to the opposite side.
Part 2 – Exponents & Roots Simplify the expression. Eliminate any negative exponents.
6) (−3𝑥2𝑦−4)3 7) (3𝑥3)
−1
6𝑥−2
8) √7 + √28 9) √7 ∙ √28
10) √48
√3 11)
(𝑥2𝑦3)4
(𝑥𝑦4)−3
𝑥2𝑦
12) √16𝑥84 13)
(9𝑠𝑡)3
2⁄
(27𝑠3𝑡−4)2
3⁄
Properties of Exponents
𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛 Ex: 𝑥5 ∙ 𝑥2 = 𝑥7 𝑎𝑚
𝑎𝑛 = 𝑎𝑚−𝑛 Ex: 𝑥8
𝑥5 = 𝑥3 𝑎0 = 1 𝑎 ≠ 0
(𝑎𝑚)𝑛 = 𝑎𝑚∙𝑛 Ex: (𝑥5)2 = 𝑥10 (𝑎
𝑏)
𝑚=
𝑎𝑚
𝑏𝑚 Ex: (2
𝑥)
3=
8
𝑥3 𝑎−𝑛 =1
𝑎𝑛 Ex: 𝑥−2
1=
1
𝑥2
(𝑎𝑏)𝑚 = 𝑎𝑚𝑏𝑚 Ex: (4𝑥𝑦2)3 = 64𝑥3𝑦6 1
𝑎−𝑛 = 𝑎𝑛 Ex: 1
𝑥−2 = 𝑥2
Properties of nth Roots
√𝑎𝑛
= 𝑏 means 𝑏𝑛 = 𝑎 Definition of the nth root: 𝑎1
𝑛 = √𝑎𝑛
1) √𝑎𝑏𝑛
= √𝑎𝑛
∙ √𝑏𝑛
Ex: √−8 ∙ 273
= √−83
∙ √273
= −2(3) = −6
2) √𝑎
𝑏
𝑛=
√𝑎𝑛
√𝑏𝑛 Ex: √
16
81
4=
√164
√814 =
2
3
3) √ √𝑎𝑛𝑚
= √𝑎𝑚𝑛
Ex: √√7293
= √7296
= 3
4) √𝑎𝑛𝑛= 𝑎 if n is odd Ex: √(−5)33
= −5
5) √𝑎𝑛𝑛= |𝑎| if n is even Ex: √(−3)44
= |−3| = 3
Part 3 – Factoring & Solving Quadratic Equations Factor completely each expression. 14) 𝑥2 + 8𝑥 + 7 15) 𝑥2 + 2𝑥 − 24 16) 2𝑥2 − 7𝑥 + 3 17) 4𝑥2 + 27𝑥 + 35 18) −28𝑦3 + 7𝑦2 19) 𝑥3 − 2𝑥2 − 9𝑥 + 18 20) 𝑥3 + 8 21) 4𝑥2 − 121 22) 8𝑎4 + 27𝑎𝑏3 Solve the equation. 23) −3𝑥2 − 5𝑥 + 12 = 0 24) 4𝑥2 + 12𝑥 + 9 = 0 25) 𝑥2 + 2𝑥 + 2 = 0
Factoring Methods
GCF
Guess and check
Grouping
Difference of two squares Sum/difference of cubes
Part 4 – Rational Expressions
Multiplying
𝐴
𝐵∙
𝐶
𝐷=
𝐴𝐶
𝐵𝐷
Multiply numerators and multiply denominators Simplifying
𝐴𝐶
𝐵𝐶=
𝐴
𝐵
Factor both numerator and denominator and “cancel” common factors.
Dividing
𝐴
𝐵÷
𝐶
𝐷=
𝐴
𝐵∙
𝐷
𝐶
Keep the first fraction, flip the second and multiply
Adding Like denominators:
𝐴
𝐶+
𝐵
𝐶=
𝐴 + 𝐵
𝐶
Unlike denominators:
𝐴
𝐶+
𝐵
𝐷=
𝐴𝐷 + 𝐵𝐶
𝐶𝐷
Example:
Add 3
𝑥−1+
𝑥
𝑥+2=
3(𝑥+2)+𝑥(𝑥−1)
(𝑥−1)(𝑥+2)=
3𝑥+6+𝑥2−𝑥
(𝑥−1)(𝑥+2)=
𝑥2+2𝑥+6
(𝑥−1)(𝑥+2)
Simplify the expression.
26) 3(𝑥+2)(𝑥−1)
6(𝑥−1)2 27)
𝑥2−𝑥−12
𝑥2+5𝑥+6
Perform the indicated operation and then simplify completely.
28) 4𝑥
𝑥2−4∙
𝑥+2
16𝑥 29)
1
𝑥+5+
2
𝑥−3 30)
4𝑦2−9
2𝑦2+9𝑦−18÷
2𝑦2+𝑦−3
𝑦2+5𝑦−6
31) 𝑥3
𝑥+1𝑥
𝑥2+2𝑥+1
32) 𝑥
𝑥−1+1
𝑥+2
𝑥
Part 5 – Inequalities Solve each inequality. Graph the solution. 33) 𝑥 + 2 ≥ 15 34) ) 5 − 3𝑥 ≤ 35 35) −3 < 2𝑥 + 1 ≤ 3 Part 6 – Functions & Graphs Determine if the graph represents a function. 36) 37) 38) 39) Sketch the graph of the each function. If you need a reminder, use your graphing calculator to help remember transformations of functions. Keep in mind, you need to be able to graph functions without a graphing calculator. 40) 𝑓(𝑥) = 2𝑥 + 2 41) 𝑓(𝑥) = (𝑥 + 2)2 − 2 42) 𝑓(𝑥) = |𝑥| + 3
43) 𝑓(𝑥) = √𝑥 − 3 + 1 44) 𝑓(𝑥) = (𝑥 + 1)3 + 1 45) 𝑓(𝑥) =1
𝑥
Combining Functions & Compositions of Functions
Let 𝑓(𝑥) =1
𝑥−2 and 𝑔(𝑥) = √𝑥
Combining Functions
𝑓(𝑥) + 𝑔(𝑥) Ex: (𝑓 + 𝑔)𝑥 =1
𝑥−2+ √𝑥
𝑓(𝑥) − 𝑔(𝑥) Ex: (𝑓 − 𝑔)𝑥 =1
𝑥−2− √𝑥
Given two functions f and g, the composite function, 𝑓 ∘ 𝑔, (also called the composition of f and g) is defined by
(𝑓 ∘ 𝑔)(𝑥) = 𝑓(𝑔(𝑥))
Find 𝑓 + 𝑔 𝑎𝑛𝑑 𝑓 − 𝑔
46) 𝑓(𝑥) = 𝑥 − 3, 𝑔(𝑥) = 𝑥2 47) 𝑓(𝑥) =2
𝑥 , 𝑔(𝑥) =
4
𝑥+4
48) Given 𝑓(𝑥) = 6𝑥 − 5 and 𝑔(𝑥) =𝑥
2 find: 49) Given 𝑓(𝑥) = 𝑥2 and 𝑔(𝑥) = √𝑥 − 3
find: a) 𝑓 ∘ 𝑔 a) 𝑓 ∘ 𝑔
b) 𝑔 ∘ 𝑓 b) 𝑔 ∘ 𝑓
c) 𝑓 ∘ 𝑓 c) 𝑔 ∘ 𝑔
Part 7 – Polynomial Functions Divide the polynomials using long division 50) 𝑃(𝑥) = 8𝑥4 + 6𝑥2 − 3𝑥 + 1 51) 𝑃(𝑥) = 2𝑥3 − 3𝑥2 − 2𝑥 𝐷(𝑥) = 2𝑥2 − 𝑥 + 2 𝐷(𝑥) = 2𝑥 − 3 Divide the polynomials using synthetic division. 52) 𝑓(𝑥) = 𝑥3 + 2𝑥2 + 2𝑥 + 1, 𝑔(𝑥) = 𝑥 + 2 53) ℎ(𝑥) = 2𝑥4 + 3𝑥3 − 12, 𝑗(𝑥) = 𝑥 + 4
54) For the graph pictured at the right:
a) Describe the end behavior
b) Determine whether it represents an odd-degree function or an even-degree function
c) State the number of real zeros
Polynomial Division
Long Division EX: 2𝑥3 − 5𝑥2 + 𝑥 − 7 ÷ 𝑥 − 3 Synthetic Division
Quotient:
2𝑥2 + 𝑥 + 4 +5
𝑥−3
2𝑥2 + 𝑥 + 4 +5
𝑥 − 3
Part 8- Right Triangles & Trigonometry Find the value of the trig function expressed as a fraction and as decimal to the nearest hundredth. Find the value of the angle (𝜃) to the nearest degree. Show all work for set up of ratios and trig equations. 55) 56)
sin 𝜃 = _________ sin 𝜃 = _________
cos 𝜃 = _________ cos 𝜃 = _________
tan 𝜃 = _________ tan 𝜃 = _________
𝜃 = _________ 𝜃 = _________
Solve for the value of x.
57) 58) 59)
cos 𝐴 =𝑎𝑑𝑗
ℎ𝑦𝑝=
𝑏
𝑐 cos−1 (
𝑏
𝑐) = ∠𝐴
tan 𝐴 =𝑜𝑝𝑝
𝑎𝑑𝑗=
𝑎
𝑏 tan−1 (
𝑎
𝑏) = ∠𝐴
For the right triangle pictured:
SOHCAHTOA
Pythagorean Theorem sin 𝐴 =𝑜𝑝𝑝
ℎ𝑦𝑝=
𝑎
𝑐 sin−1 (
𝑎
𝑐) = ∠𝐴
𝑎2 + 𝑏2 = 𝑐2
x x
x