10
Lyman Memorial High School Honors Pre-Calculus Prerequisite Packet Fall 2017 Name:__________________________

Lyman Memorial High School Honors Pre-Calculus ... Pre-Calc Summer... · Lyman Memorial High School Honors Pre-Calculus Prerequisite Packet ... and practiced throughout the summer

Embed Size (px)

Citation preview

Page 1: Lyman Memorial High School Honors Pre-Calculus ... Pre-Calc Summer... · Lyman Memorial High School Honors Pre-Calculus Prerequisite Packet ... and practiced throughout the summer

Lyman Memorial High School

Honors Pre-Calculus Prerequisite Packet

Fall 2017

Name:__________________________

Page 2: Lyman Memorial High School Honors Pre-Calculus ... Pre-Calc Summer... · Lyman Memorial High School Honors Pre-Calculus Prerequisite Packet ... and practiced throughout the summer

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

Page 3: Lyman Memorial High School Honors Pre-Calculus ... Pre-Calc Summer... · Lyman Memorial High School Honors Pre-Calculus Prerequisite Packet ... and practiced throughout the summer

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.

Page 4: Lyman Memorial High School Honors Pre-Calculus ... Pre-Calc Summer... · Lyman Memorial High School Honors Pre-Calculus Prerequisite Packet ... and practiced throughout the summer

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

Page 5: Lyman Memorial High School Honors Pre-Calculus ... Pre-Calc Summer... · Lyman Memorial High School Honors Pre-Calculus Prerequisite Packet ... and practiced throughout the summer

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

Page 6: Lyman Memorial High School Honors Pre-Calculus ... Pre-Calc Summer... · Lyman Memorial High School Honors Pre-Calculus Prerequisite Packet ... and practiced throughout the summer

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

Page 7: Lyman Memorial High School Honors Pre-Calculus ... Pre-Calc Summer... · Lyman Memorial High School Honors Pre-Calculus Prerequisite Packet ... and practiced throughout the summer

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

Page 8: Lyman Memorial High School Honors Pre-Calculus ... Pre-Calc Summer... · Lyman Memorial High School Honors Pre-Calculus Prerequisite Packet ... and practiced throughout the summer

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

Page 9: Lyman Memorial High School Honors Pre-Calculus ... Pre-Calc Summer... · Lyman Memorial High School Honors Pre-Calculus Prerequisite Packet ... and practiced throughout the summer

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

Page 10: Lyman Memorial High School Honors Pre-Calculus ... Pre-Calc Summer... · Lyman Memorial High School Honors Pre-Calculus Prerequisite Packet ... and practiced throughout the summer

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