Math Algebra 2 Week of June 1 Grades 9_ 10_ and 11...Math – Algebra 2 – Week of June 1 Assignments NOTE • The study guide for the final exam, a supplemental problems packet,

Math – Algebra 2 – Week of June 1

Assignments

NOTE

• The study guide for the final exam, a supplemental problems packet, and a formula sheet are included at the bottom of this packet. The study guide and supplemental problems packet will be due by the end of the day on Wednesday, June 17th. Specifically: o All: Complete the Study Guide o H: Complete the odd-numbered problems from the

Supplemental Problems set o HH: Complete all problems from the Supplemental Problems

set

• REMEMBER: The last day for Quarter 4 late work will be Wednesday, June 17th.

Monday, June 8

• All: Watch video on Lesson 116 on Microsoft Teams

• All: Complete 10 min. each on IXL Algebra 2 skills EE.6 (Find the equation of a regression line) and EE.7 (Interpret regression lines)

• All: Complete Lesson 116: 1-30 from textbook

Tuesday, June 9

• All: Watch video on Lesson 117 on Microsoft Teams

• All: All: Complete 15 min. on IXL Algebra 2 skill E.16 (Solve a non-linear system of equations)

• CP: Complete Lesson 117 worksheet (below)

• H and HH: Complete Lesson 117: 1-30 from textbook

Wednesday, June 10

• All: Watch video on Lesson 118 on Microsoft Teams

• All: Complete the Lesson 118 lesson practice (below)

• CP: Complete Lesson 118 worksheet (below)

• H and HH: Complete Lesson 118: 1-30 from textbook

Thursday, June 11

• All: Work on the final exam study guide and supplemental problem set. Again, they are due by the end of the day on Wednesday, June 17th.

• All: Work on any Quarter 4 missing work. It is also due by the end of the day on Wednesday, June 17th.

Friday, June 12

• All: Work on the final exam study guide and supplemental problem set. Again, they are due by the end of the day on Wednesday, June 17th.

• All: Work on any Quarter 4 missing work. It is also due by the end of the day on Wednesday, June 17th.

Instructions

• You must submit the homework to your teacher by Turnitin by the end of the day two days after it was assigned. For example, the homework for Monday is due by the end of Wednesday. The homework for Thursday and Friday are due by the end of the following Monday. To submit, you may EITHER:

o Take pictures of your work. Put all pictures into a single Word document. Save the Word document as a PDF. Submit on Turnitin. OR, you may:

o Scan your completed work as a PDF. Upload the PDF to Turnitin.

• Write legibly.

• Each IXL assignment will be worth a participation grade of 10 points. Participation grades will be posted to Power School. Each day’s homework/worksheet assignment will be worth a homework grade of 10 points. Homework grades will be posted to Power School.

• IXL assignments are not uploaded to Turnitin. Notes copied into/taken in notebook do not need to be photographed and submitted.

• Collaboration is not allowed. Collaboration: To work jointly with others or together especially in an intellectual endeavor. When collaboration takes place, all students must demonstrate understanding of the new material.

Ms. Reynolds– Algebra 2 Homework

Name: ____________________________________________ Date: _________________________

Lesson # 117 – Solving Systems of Nonlinear Equations

*In addition to the following problems, please also complete #’s {2, 4, 6, 7, 9, 16, 18, 21, 25,

26, 28, 30} from the textbook.

(1) Solve the system of equations using the method of substitution:

{𝑥2 − 𝑦2 = 48𝑥 = 7𝑦

(2) Solve the system of equations using the method of substitution:

{𝑦 + 2 = 2𝑥

𝑥𝑦 = 24

(3) Solve the system of equations using the method of substitution:

{5𝑦 = 3𝑥𝑥𝑦 = 15

(4) Solve the system of equations using the method of elimination:

{𝑥2 − 𝑦2 = 7

𝑥2 + 𝑦2 = 25

(5) Solve the system of equations using the method of elimination:

{𝑥2 + 𝑦2 = 16

𝑥2 − 2𝑦2 = 1

*In addition to the following problems, please also complete #’s {2, 4, 6, 7, 9, 16, 18, 21, 25,

26, 28, 30} from the textbook.

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

1 | P a g e

Algebra 2 Lesson 118 Lesson Practice

1: A sales person for a new lotion gave samples of the lotion to four dermatologists and asked

them if they would recommend it to their patients. Three of the dermatologists said they would.

The sales person wrote a brochure stating that 3 out of 4 dermatologists recommend the lotion.

Why is the claim misleading?

2: A student wants to know the average height difference between boys and girls in the eleventh

grade. In 11th-grade homerooms, the student walks up and down the rows, asking other students

to state their height, in inches. How can the results of this survey be misleading?

3: Explain how the graph could be misleading?

4: A study showed that individuals who visit a dentist twice a year tend to make more money

than individuals who visit less than twice a year. Can it be concluded that visiting a dentist

causes a person to make more money? Explain.

Ms. Reynolds– Algebra 2 Homework

Name: ____________________________________________ Date: _________________________

Lesson # 118 – Recognizing Misleading Data

*In addition to the following problems, please also complete #’s {1, 4, 7, 10, 13, 15, 17, 21,

22, 29, 30} from the textbook.

(1) A sales person for a new perfume gives samples of the perfume to four dermatologists and asks if they would recommend the product to their patients. Three of the dermatologists say that they would. The sales person wrote a brochure stating that 3 out of 4 dermatologists would recommend the perfume. Why is this claim misleading?

(2) The manager of an apartment building posts the following survey in every apartments’ mailbox: “Next month, maintenance workers must enter homes for an annual inspection. You must be home during the inspection. Your options are (1) weekdays before noon ; (2) weekdays between noon and 5pm ; (3) weekdays between 5pm and 9pm. Which option is most convenient for you?” How can the results of the survey be misleading?

(3) Explain why the presentation of the graph below can be misleading. (4) Explain why the presentation of the graph below can be misleading.

*In addition to the following problems, please also complete #’s {1, 4, 7, 10, 13, 15, 17, 21, 22, 29, 30} from the textbook. _____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

____________________________________________________________________________________

Algebra 2 Formula Packet

• Slope Intercept Form of a Line: 𝑦 = 𝑚𝑥 + 𝑏

• Point Slope Form of a Line: 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)

• Standard Form of a Quadratic Equation: 𝑦 = 𝐴𝑥2 + 𝐵𝑥 + 𝐶

• Vertex Form of a Quadratic Equation: 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 ;𝑤ℎ𝑒𝑟𝑒 (ℎ, 𝑘) = 𝑣𝑒𝑟𝑡𝑒𝑥

• Distance Formula: 𝑑 = √(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2

• Pythagorean Theorem:

𝑎2 + 𝑏2 = 𝑐2 ; 𝑤ℎ𝑒𝑟𝑒 𝑐 𝑖𝑠 𝑡ℎ𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑜𝑓 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒

• Permutation Formula: 𝑛𝑃𝑟 =𝑛!

(𝑛−𝑟)!

• Combination Formula: 𝑛𝐶𝑟 =𝑛!

(𝑛−𝑟)!𝑟!

• Determinant of 𝐴 = [𝑎 𝑏𝑐 𝑑

]: 𝑎𝑑 − 𝑏𝑐

• Inverse of 𝐴 = [𝑎 𝑏𝑐 𝑑

]: 1

det [𝐴][𝑑 −𝑏−𝑐 𝑎

]

• Cramer’s Rule: 𝐺𝑖𝑣𝑒𝑛 {𝑎𝑥 + 𝑏𝑦 = 𝑒𝑐𝑥 + 𝑑𝑦 = 𝑓

, 𝑡ℎ𝑒𝑛 𝑥 =𝑑𝑒𝑡[

𝑒 𝑏𝑓 𝑑

]

𝑑𝑒𝑡[𝑎 𝑏𝑐 𝑑

] & 𝑦 =

𝑑𝑒𝑡[𝑎 𝑒𝑐 𝑓]

𝑑𝑒𝑡[𝑎 𝑏𝑐 𝑑

]

• Trigonometric Ratios: 𝑆𝑖𝑛(𝜃) =𝑜𝑝𝑝

ℎ𝑦𝑝; 𝐶𝑜𝑠(𝜃) =

𝑎𝑑𝑗

ℎ𝑦𝑝; 𝑇𝑎𝑛(𝜃) =

𝑜𝑝𝑝

𝑎𝑑𝑗

• Product Rule for Radicals: √𝑎𝑏𝑛

= (√𝑎𝑛

)( √𝑏𝑛

)

• 𝑸𝒖𝒂𝒅𝒓𝒂𝒕𝒊𝒄 𝑭𝒐𝒓𝒎𝒖𝒍𝒂: 𝒙 =−𝑩±√𝑩𝟐−𝟒𝑨𝑪

𝟐𝑨

• 𝑫𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒐𝒇 𝑻𝒘𝒐 𝑺𝒒𝒖𝒂𝒓𝒆𝒔: 𝒂𝟐 − 𝒃𝟐 = (𝒂 + 𝒃)(𝒂 − 𝒃)

• 𝑫𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒐𝒇 𝑻𝒘𝒐 𝑪𝒖𝒃𝒆𝒔: 𝒂𝟑 − 𝒃𝟑 = (𝒂 − 𝒃)(𝒂𝟐 + 𝒂𝒃 + 𝒃𝟐)

• 𝑺𝒖𝒎 𝒐𝒇 𝑻𝒘𝒐 𝑪𝒖𝒃𝒆𝒔: 𝒂𝟑 + 𝒃𝟑 = (𝒂 − 𝒃)(𝒂𝟐 + 𝒂𝒃 + 𝒃𝟐)

• 𝑩𝒊𝒏𝒐𝒎𝒊𝒂𝒍 𝒕𝒉𝒆𝒐𝒓𝒆𝒎: (𝒙 + 𝒚)𝒏 = 𝒏𝒄𝟎 𝒙𝒏𝒚𝟎 + 𝒏𝒄𝟏 𝒙

𝒏−𝟏𝒚𝟏 + 𝒏𝒄𝟐 𝒙𝒏−𝟐𝒚𝟐 + ⋯+

𝒏𝒄𝒏𝒙𝟎𝒚𝒏

Parent graph for 𝒚 = 𝐬𝐢𝐧𝜽

Parent graph for 𝒚 = 𝐜𝐨𝐬 𝜽

Parent graph for 𝒚 = 𝐭𝐚𝐧𝜽

Sum and Difference Trig Identities: sin(𝐴 + 𝐵) = sin 𝐴 cos𝐵 + cos 𝐴 sin𝐵 sin(𝐴 − 𝐵) = sin 𝐴 cos𝐵 − cos 𝐴 sin𝐵

cos(𝐴 + 𝐵) = cos 𝐴 cos 𝐵 − sin𝐴 sin 𝐵 cos(𝐴 − 𝐵) = cos 𝐴 cos 𝐵 + sin𝐴 sin 𝐵

tan(𝐴 + 𝐵) =tan𝐴 + tan𝐵

1 − tan𝐴 tan𝐵

tan(𝐴 − 𝐵) =tan𝐴 − tan𝐵

1 + tan𝐴 tan𝐵

Law of Sines 𝑎

sin𝐴=

𝑏

sin𝐵=

𝑐

sin 𝐶

Law of Cosines 𝑎2 = 𝑏2 + 𝑐2 − 2𝑏𝑐 cos𝐴

𝒏th term of an Arithmetic Sequence:

𝑎𝑛 = 𝑎𝑚 + (𝑛 − 𝑚)𝑑

𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑

𝒏th term of a Geometric Sequence:

𝑎𝑛 = 𝑎𝑚 ⋅ 𝑟𝑛−𝑚 𝑎𝑛 = 𝑎1 ⋅ 𝑟𝑛−1

Sum of the first 𝒏 terms of an Arithmetic Series

𝑆𝑛 =𝑛

2(𝑎1 + 𝑎𝑛) =

𝑛

2(2𝑎1 + (𝑛 − 1)𝑑)

Sum of the first 𝒏 terms of a Geometric Series

𝑆𝑛 =𝑎1(𝑟

𝑛 − 1)

𝑟 − 1

Difference of Two Squares

𝑎2 − 𝑏2 = (𝑎 + 𝑏)(𝑎 − 𝑏)

Difference of Two Cubes

𝑎3 − 𝑏3 = (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏2)

Sum of Two Cubes

𝑎3 + 𝑏3 = (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏2)

The Quadratic Formula for 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎

𝑥 =−𝑏 ± √𝑏2 − 4𝑎𝑐

2𝑎

Logarithm Properties

log(𝑎𝑏) = log(𝑎) + log(𝑏)

log (𝑎

𝑏) = log(𝑎) − log(𝑏)

log(𝑎𝑏) = 𝑏 ⋅ log(𝑎)

ln(𝑒) = 1

Equation of a Circle

(𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟2 (ℎ, 𝑘) is the center of the circle 𝑟 is the radius of the circle

Equation of a horizontal Ellipse

(𝑥 − ℎ)2

𝑎2+

(𝑦 − 𝑘)2

𝑏2= 1

(ℎ, 𝑘) is the center of the ellipse

𝑐2 = 𝑎2 − 𝑏2 eccentricity 𝑒 =𝑐

𝑎

Equation of a vertical Ellipse (𝑦 − 𝑘)2

𝑎2+

(𝑥 − ℎ)2

𝑏2= 1

Vector Algebra Dot product of two vectors

|𝑣1⃑⃑⃑⃑ |. |𝑣2⃑⃑⃑⃑ | = 𝑥1𝑥2 + 𝑦1𝑦2 Magnitude of a vector

|𝑣1⃑⃑⃑⃑ | = √𝑥12 + 𝑦1

2

|𝑣1⃑⃑⃑⃑ |. |𝑣2⃑⃑⃑⃑ | = |𝑣1⃑⃑ ⃑||𝑣2⃑⃑ ⃑| cos 𝜃

Equation of a Horizontal Hyperbola

(𝑥 − ℎ)2

𝑎2−

(𝑦 − 𝑘)2

𝑏2= 1

(ℎ, 𝑘) is the center of the hyperbola 𝑐2 = 𝑎2 + 𝑏2

eccentricity 𝑒 =𝑐

𝑎

Equation of a Vertical Hyperbola

(𝑦 − 𝑘)2

𝑎2−

(𝑥 − ℎ)2

𝑏2= 1

(ℎ, 𝑘) is the center of the hyperbola 𝑐2 = 𝑎2 + 𝑏2

eccentricity 𝑒 =𝑐

𝑎

Half angle formulas

𝑆𝑖𝑛𝜃

2= ∓√

1 − cos 𝜃

2

𝐶𝑜𝑠𝜃

2= ∓√

1 + cos 𝜃

2

𝑡𝑎𝑛 𝜃

2= ∓√

1 − cos 𝜃

1 + 𝑐𝑜𝑠 𝜃

Double angle formulas

sin 2𝜃 = 2 𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃

cos 2𝜃 = 𝑐𝑜𝑠2𝜃 − 𝑠𝑖𝑛2𝜃

cos 2𝜃 = 2𝑐𝑜𝑠2𝜃 − 1

cos 2𝜃 = 1 − 2𝑠𝑖𝑛2𝜃

tan 2𝜃 = 2 tan 𝜃

1 − 𝑡𝑎𝑛2𝜃

THE UNIT CIRCLE

1

Algebra 2 Final Exam Study Guide 2019-2020

*The following study guide is comprised of helpful information to remember while studying for the

final exam. This information includes formulas that you will need to know for the exam, as well as

example problems that are similar to problems you will be tested on. The study guide is broken

down into different Algebra 2 topics: (1) Functions, (2) Roots, (3) Inequalities, (4) Trigonometry,

(5)Complex Expressions, (6) Logarithms,(7) Factoring, (8) Sequences & Series, and (9) Conic Sections

*Functions*

• Recall that to evaluate a function 𝑓(𝑥) for a given value, “a”, you simply replace every “x” in

the expression with “a.” Sometimes the function changes depending on the value of x that

you plug in. This is referred to as a piecewise function.

Example 1:

Given that 𝑓(𝑥) = {𝑥 + 2 𝑖𝑓 𝑥 ≤ 6

𝑥2 − 1 𝑖𝑓 𝑥 > 6 calculate 𝑓(7) − 𝑓(3)

• When two functions are inverses of each other, they are symmetrical over the line 𝑦 = 𝑥.

• To find the inverse of a given function, you should switch the x and y values and then re-

solve for y

Example 2:

Given that 𝑦 = (𝑥 + 9)2 find the inverse function, 𝑓−1(𝑥)

Step 1: Switch the x and y variables in the original equation

Step 2: Use algebra to resolve for y

2

• When graphed, rational functions have asymptotes.

• The vertical asymptote occurs at any value which makes the function undefined (where the

denominator is equal to 0)

• The horizontal asymptote can be found by comparing the degree of the numerator to the

degree of the denominator

• If:

{

𝑑𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 < 𝑑𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟… . . 𝑡ℎ𝑒𝑛 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑎𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒 𝑖𝑠 𝑦 = 0

𝑑𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 = 𝑑𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟… . 𝑡ℎ𝑒𝑛 𝐻. 𝐴.= 𝑑𝑖𝑣𝑖𝑠𝑖𝑜𝑛 𝑜𝑓 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑠

𝑑𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 = 𝑑𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 + 1… . 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒𝑟𝑒′𝑠 𝑎 𝑠𝑙𝑎𝑛𝑡 𝑎𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒

• To calculate the slant asymptote, divide the denominator into the numerator and throw out

the remainder

Example 3:

Determine all asymptotes of the following rational functions:

𝑓(𝑥) =3𝑥+4

4𝑥+2

𝑔(𝑥) =𝑥2+5𝑥+3

𝑥−1

• The “end behavior” of a function describes what happens as the x values of the functions get

very large or very small. This is based off of whether the degree of the function is ODD or EVEN

and whether the leading coefficient is POSITIVE or NEGATIVE

POSITIVE NEGATIVE

ODD Left Down / Right Up

Left Up / Right Down

EVEN Both Up Both Down

3

• Recall that functions can be transformed (shifted, stretched, or reflected)

• The following is a table reminding you about all of the different transformations:

𝑓(𝑥) + 𝑘 Vertical shift up k units

𝑓(𝑥) − 𝑘 Vertical shift down k units

𝑓(𝑥 + 𝑘) Horizontal shift left k units

𝑓(𝑥 − 𝑘) Horizontal shift right k units

𝑐𝑓(𝑥) Vertical stretch by a factor of c

𝑓(𝑐𝑥) Horizontal stretch by a factor of 1/c

−𝑓(𝑥) Vertical reflection

𝑓(−𝑥) Horizontal reflection

Example 4:

Given the parent graphs 𝑓(𝑥) = √𝑥 and 𝑓(𝑥) = 𝑥2, sketch a graph of the following transformation

functions

(a) 𝑔(𝑥) = (𝑥 − 2)2 + 1

(b) 𝑔(𝑥) = √𝑥 + 4 − 3

(c) 𝑔(𝑥) = −√𝑥 − 2

4

*Roots*

• The word “root” in mathematics is a synonym for “zero” or “x-intercept”. In other words, it is

where the graph crosses the X-axis

• Oftentimes, the easiest way to find the root of a function is to set the function equal to zero and

then factor it

Example 5:

Find all of the roots of the function 𝑓(𝑥) = 4𝑥3 + 24𝑥2 + 32𝑥

Step 1: Set the function equal to 0

Step 2: Factor out the GCF from the polynomial function

Step 3: Factor the quadratic function inside the parentheses into two binomials

Step 4: Set each function equal to 0 separately and solve for x

• If you are given a function 𝑓(𝑥) and 𝑥 = 𝑎 is a root then 𝑓(𝑥) will have a zero remainder when

divided by (𝑥 − 𝑎)

Example 6:

Use synthetic division to determine if 𝑓(𝑥) = 𝑥3 − 𝑥2 − 17𝑥 − 15 has a zero remainder when

divided by (𝑥 − 5)

5

*Inequalities*

• To solve a rational or quadratic inequality you must first calculate all values that make the

function equal to zero as well as all values that make the function undefined.

• Once these values are determined, put them on a number line and test each interval to

determine where the function is negative or positive

• Lastly, determine which regions (positive or negative) are part of the solution set based on the

inequality symbol (for example, greater than 0 is positive and less than 0 is negative). Make sure

to write your answer as an inequality!

Example 7:

Solve the quadratic inequality 𝑥2 − 3𝑥 − 28 ≤ 0

Step 1: Factor the quadratic expression into two binomials and identify the roots

Step 2: Put the roots on a number line in numerical order

Step 3: Test each interval of the number line and label it as positive or negative

Step 4: Determine which intervals satisfy the original inequality

Step 5: Write your answer as an inequality

Example 8: Solve the rational inequality 𝑥+5

𝑥−8> 0

Step 1: Identify the roots of the numerator and denominator

Step 2: Put the roots on a number line in numerical order

Step 3: Test each interval of the number line and label it as positive or negative

Step 4: Determine which intervals satisfy the original inequality

Step 5: Write your answer as an inequality

6

*Trigonometry*

• Trigonometry can be used when solving for side lengths or angles of a triangle.

•

If:

{

𝑖𝑡 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒… . . 𝑦𝑜𝑢 𝑐𝑎𝑛 𝑢𝑠𝑒 𝑆𝑜ℎ − 𝐶𝑎ℎ − 𝑇𝑜𝑎

𝑦𝑜𝑢 𝑎𝑟𝑒 𝑔𝑖𝑣𝑒𝑛 𝑎𝑛 𝑎𝑛𝑔𝑙𝑒 𝑎𝑛𝑑 𝑖𝑡𝑠 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 𝑙𝑒𝑛𝑔𝑡ℎ…𝑦𝑜𝑢 𝑐𝑎𝑛 𝑢𝑠𝑒 𝐿𝑎𝑤 𝑜𝑓 𝑆𝑖𝑛𝑒𝑠

𝑦𝑜𝑢 𝑎𝑟𝑒 𝑔𝑖𝑣𝑒𝑛 𝑜𝑛𝑙𝑦 𝑡ℎ𝑒 𝑠𝑖𝑑𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒… . 𝑦𝑜𝑢 𝑐𝑎𝑛 𝑢𝑠𝑒 𝐿𝑎𝑤 𝑜𝑓 𝐶𝑜𝑠𝑖𝑛𝑒𝑠

Examples 9:

Solve for the missing angle:

Step 1: Write a trigonometric equation using SOH-CAH-TOA

Step 2: Solve for x using algebra

Example 10 Solve for the missing side:

Step 1: Use the given information to set up a proportion using the Law of Sines: 𝑎

sin (𝐴)=

𝑏

sin (𝐵)

Step 2: Cross multiply and solve for the missing variable

7

• A trigonometric equation is an equation involving a trigonometric function in which your goal is

to solve for the angle 𝜃

• To do this, you should isolate the trig function and then take the inverse of both sides

• You will normally have 2 answers for 𝜃 but the calculator will only give you one of them so you

must use reference angles to figure out the other

Example 11:

Solve for 𝜃: 6𝑠𝑖𝑛𝜃 − 3 = 0 where 0 ≤ 𝜃 ≤ 2𝜋

Step 1: Isolate 𝑠𝑖𝑛𝜃 on the left hand side of the equation

Step 2: Use a copy of the unit circle to identify which angles satisfy the equation (remember that sine

refers to the y-value of the point!)

• To find the “exact” value of a trigonometric function, you sometimes have to use Sum and

Difference formulas. These WILL be provided for you on the exam, as well as a unit circle.

Example 12: Find the exact value of 𝑆𝑖𝑛(165°) using a Sum or Difference formula

Step 1: Re-write 165 as the sum of two angles found on the unit circle

Step 2: Plug these angles into the formula: 𝑆𝑖𝑛(𝐴 + 𝐵) = 𝑆𝑖𝑛(𝐴)𝐶𝑜𝑠(𝐵) + 𝑆𝑖𝑛(𝐵)𝐶𝑜𝑠(𝐴)

Step 3: Identify these values from the unit circle and substitute them into the formula

Step 4: Simplify your result as far as possible

8

• All parent graphs of trig functions will be provided for you on the test. You have to transform

them according to the given function. Recall that the general form of transformations is:

𝑦 = 𝐴𝑠𝑖𝑛(𝐵𝑥) + 𝐶 𝑦 = 𝐴𝑐𝑜𝑠(𝐵𝑥) + 𝐶 𝑦 = 𝐴𝑡𝑎𝑛(𝐵𝑥) + 𝐶

• A = amplitude (vertical stretch of the graph)

• B determines the period of the function (how many radians until the cycle repeats)

• For Sine and Cosine functions, the Period = 2𝜋

𝐵 ; for Tangent functions the Period =

𝜋

𝐵

• The C value added or subtracted at the end of the function is a vertical shift up or down

Example 13:

Graph the function 𝑦 = 5 cos(2𝑥) ; identify the amplitude and period

*Complex Expressions*

• Recall that √−1 = 𝑖

• Thus: 𝑖1 = 𝑖 ; 𝑖2 = −1 ; 𝑖3 = −𝑖 ; 𝑖4 = 1

• To calculate a power of 𝑖 higher than 4, you should divide the exponent by 4 and calculate the

remainder. This remainder will match up with one of the original 4 powers of 𝑖

Example 14:

Simplify 5𝑖18 + 6𝑖27

Step 1: Divide each of the exponents by 4 and calculate their remainders

Step 2: Match up the remainders with one of the original 4 powers of 𝑖

Step 3: Substitute these powers of 𝑖 into the original expression and simplify the answer as far as

possible

9

• It is technically considered mathematically incorrect to leave an 𝑖 in the denominator of an

expression (just like a square root)

• In order to get the 𝑖 out of the denominator, you must multiply the numerator and denominator

by a form of 1 (if the denominator is a binomial, you multiply by a conjugate)

Example 15:

Simplify the complex expression 2

6+3𝑖 and write the answer in the form 𝑎 + 𝑏𝑖

Step 1: Multiply the numerator and denominator by the conjugate of the denominator (6 − 3𝑖)

Step 2: Use the distributive property to simplify the numerator and use the FOIL method to expand

out the denominator

Step 3: Simplify your result as far as possible

*Logarithms*

• Logarithms and Exponential Functions are inverses of each other

• Recall that saying 𝑎𝑥 = 𝑦 is the same thing as saying 𝑙𝑜𝑔𝑎(𝑦) = 𝑥

• There are two special logarithms:

1. Common Log: 𝑙𝑜𝑔10(𝑥) = log (𝑥)

2. Natural Log: 𝑙𝑜𝑔𝑒(𝑥) = ln (𝑥)

• There are also a number of logarithm properties

1. log(𝐴𝐵) = log(𝐴) + log (𝐵)

2. log (𝐴

𝐵) = log(𝐴) − log (𝐵)

3. log(𝐴𝐵) = 𝐵𝑙𝑜𝑔(𝐴)

4. ln(𝑒) = 1

10

Example 16:

Use the logarithm properties to expand ln (𝑥6𝑒3

𝑤5 ) as far as possible

Example 17:

Use the logarithm properties to solve the logarithmic equation: 𝑙𝑜𝑔2(𝑥 − 7) = 5

Step 1: Convert the logarithmic equation into an exponential equation

Step 2: Use algebra to solve for x

Example 18:

Solve for x: 10004𝑥 = 100 round your answer to the nearest tenth

Step 1: Take the log of both sides of the equation

Step 2: Use log properties to re-write the left hand side of the equation

Step 3: Solve the resulting equation for x

11

*Factoring*

• You will be required to factor a few polynomials using a number of methods

• These methods include factoring basic quadratics, difference of two squares, and factoring by

grouping

• Difference of Two Squares: 𝑎2 − 𝑏2 = (𝑎 + 𝑏)(𝑎 − 𝑏)

Example 19:

Factor the following expression: 64𝑤2 − 49𝑦2

Step 1: Write the expression in the form 𝑎2 − 𝑏2

Step 2: Use the difference of two squares formula to factor the expression

Example 20: Factor the following expression: 2𝑥3 + 𝑥2 − 32𝑥 − 16

Step 1: Split the expression down the middle so that there are two terms on the left and two terms on

the right

Step 2: Factor out the GCF from each side

Step 3: Factor out the common factor between the two sides

Step 4: Continue factoring any factor as far as possible

12

*Sequences & Series*

• An arithmetic sequence is a sequence of numbers wherein consecutive numbers are separated

by a common difference

• A geometric sequence is a sequence of numbers wherein consecutive numbers are separated by

a common ratio

Ex of Arithmetic Sequence: {4,7,10,13,16……} because d = 3

Ex of Geometric Sequence: {3,12,48,144…..} because r = 4

• Formula for “n”th term

1. Arithmetic Sequence: 𝑎𝑛 = 𝑎1 + (𝑛 − 1)(𝑑)

2. Geometric Sequence: 𝑎𝑛 = 𝑎1(𝑟)𝑛−1

Example 21:

Given the arithmetic sequence {5,12,19,26…..}, first identify the common difference and then find the

214th term.

Example 22:

Given the geometric sequence {4,12,36,108…..}, first identify the common ratio and then find the

13th term.

13

• To sum up the first “n” terms of an arithmetic series we can use one of the following formula:

𝑆𝑛 =𝑛

2(𝑎1 + 𝑎𝑛)

Example 23:

Evaluate the sum of the arithmetic series:

∑3𝑘 + 2

35

𝑘=1

• To sum up the first “n” terms of an geometric series we can use the following formula:

𝑆𝑛 =𝑎1(𝑟

𝑛−1)

𝑟−1

Example 24:

Given the following geometric series {3 + 12 +48 +…….} calculate the sum of the first 11 terms

14

*Conic Sections*

• Conic Sections are parabolas, circle, ellipses, or hyperbolas

• You will be required to identify a conic section given the equation

• You will be required to write the equation for a conic section given relevant information

• You will be required to graph a conic section given relevant information

• Equation for a Parabola: 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 ; 𝑤ℎ𝑒𝑟𝑒 (ℎ, 𝑘 ) = 𝑣𝑒𝑟𝑡𝑒𝑥

• Equation for a Circle: (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟2; 𝑤ℎ𝑒𝑟𝑒 (ℎ, 𝑘) = 𝑣𝑒𝑟𝑡𝑒𝑥 & 𝑟 = 𝑟𝑎𝑑𝑖𝑢𝑠

• Equation for an Ellipse: (𝑥−ℎ)2

𝑎2+(𝑦−𝑘)2

𝑏2= 1 ; 𝑐2 = 𝑎2 − 𝑏2; 𝑒 =

𝑐

𝑎

• Equation for a Hyperbola: (𝑥−ℎ)2

𝑎2−(𝑦−𝑘)2

𝑏2= 1 ; 𝑐2 = 𝑎2 + 𝑏2; 𝑒 =

𝑐

𝑎

Example 25:

Write the equation of a circle that has a center of (−4,6) and passes through the point (2, 11)

Step 1: Use the distance formula 𝑑 = √(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)

2 to calculate the distance between

the two given points. Label this distance as the radius

Step 2: Plug all known information into the equation of a circle

15

Example 26:

Graph the following ellipse on a coordinate plane and calculate the eccentricity:

(𝑥+2)2

9+(𝑦−1)2

16= 1

Step 1: Plot the center of the ellipse on the coordinate plane

Step 2: Identify whether this is a vertical or horizontal ellipse and plot four more points using the

values of a & b, starting from the center

Step 3: Connect these four points using the shape of an ellipse

16

Example 27: Graph the following hyperbola on a coordinate plane and calculate the eccentricity

𝑥2

25−(𝑦+1)2

9= 1

Step 1: Plot the center of the hyperbola on the coordinate plane

Step 2: Identify whether this is a vertical or horizontal hyperbola and plot four more points using the

values of a & b, starting from the center

Step 3: Connect these four points using a dashed rectangle

Step 4: Draw diagonal asymptotes through the corners of the rectangular box which cross at the center

Step 5: Draw the hyperbola within the asymptotes

Algebra 2 Final Exam 2019 – 2020 – Supplemental Problems

The following are supplementary problems to help you study for the Algebra 2 Final Exam.

These problems are broken down in categories as they were on the study guide and they

are good representative samples of the types of problems you will be tested on.

Functions

1. Calculate the inverse of the function:

𝑓(𝑥) = √𝑥

6+ 1

4

2. Given the function 𝑓(𝑥) = 9𝑥 − 𝑥3 + 2𝑥5 − 3𝑥6 + 3𝑥4 + 7 + 2𝑥, determine the degree of the

function, the leading coefficient, and describe the end behavior.

3. Given the piecewise function below, calculate 𝑓(5) + 𝑓(−3) − 𝑓(−4).

𝑓(𝑥) = {2𝑥 + 5, if 𝑥 ≤ −43 − 𝑥, if − 4 < 𝑥 ≤ 0

𝑥2, if 𝑥 > 0

4. Find all asymptotes and holes for the given rational function:

𝑓(𝑥) =4𝑥2 + 2𝑥 − 1

𝑥 − 3

5. Find all asymptotes and holes for the given rational function:

𝑓(𝑥) =2𝑥2 − 5𝑥 − 3

𝑥2 + 𝑥 − 12

6. Find the inverse of 𝑦 = √2𝑥 − 1. Then, graph the original function and the inverse on the provided

coordinate grid.

Zeros of Functions / Factoring

7. Find all zeros of the function 𝑓(𝑥) = 𝑥3 + 3𝑥2 − 6𝑥 − 8.

8. Find all zeros of the function 𝑔(𝑥) = 𝑥3 − 4𝑥2 − 2𝑥 + 20.

O

2 4 6 8–2–4–6–8 x

2

4

6

8

–2

–4

–6

–8

y

9. Determine the remainder when the function 𝑃(𝑥) = 3𝑥3 + 5𝑥 − 2 is divided by (𝑥 + 4).

10. Find all zeros of the function ℎ(𝑥) = 4𝑥3 + 12𝑥2 − 𝑥 − 3.

11. Find all roots of the equation 2√5𝑥 − 4 − 10 = 0.

12. Factor using a difference of squares: (𝑥 + 2)2 − 52

13. Factor using a sum of cubes: 1000343

Inequalities

14. Solve the inequality algebraically: 𝑥2 − 8𝑥 − 20 < 0

15. Solve the inequality algebraically: 𝑥2 + 2𝑥2 − 16𝑥 − 32 ≥ 0

16. Graph the solution set to the quadratic inequality: 𝑥2 + 6𝑥 + 8 < 𝑦

O

2 4 6 8–2–4–6–8 x

2

4

6

8

–2

–4

–6

–8

y

17. Solve the inequality algebraically:

𝑥2 + 7𝑥 − 8

𝑥 − 3≥ 0

Trigonometry

18. Solve for the given angle in the following triangle:

19. Find the length of 𝑥:

20. Find the length of 𝑦:

21. Solve for 𝜃 using the unit circle: 8 cos 𝜃 + 4 = 0; 0° ≤ 𝜃 ≤ 360°

22. Solve for 𝜃 using the unit circle: 2 sin2 𝜃 − 1 = 0; 0 ≤ 𝜃 ≤ 2𝜋

23. Solve for 𝜃: 6 cos2 𝜃 + cos 𝜃 − 2 = 0; 0 ≤ 𝜃 ≤ 2𝜋

24. Find 𝑐 and 𝑑.

25. In quadrilateral 𝐴𝐵𝐶𝐷, ∠𝐴 and ∠𝐶 are right angles, 𝑚∠𝐵 = 75°, and 𝐴𝐵 = 𝐴𝐷 = 5. Find the

length 𝑧. [Hint: Start by drawing 𝐵𝐷 to split the quadrilateral into two triangles.]

26. Graph one period of the following trigonometric function. Identify the period and amplitude.

𝑦 = 4 sin(2𝜃) − 1

27. Graph one period of the following trigonometric function. Identify the period of the function.

𝑦 = tan (𝜃

2) + 3

28. Find the exact value of sin (7𝜋

12) using sum or difference identities.

29. If sin 𝜃 =8

17 and 90° < 𝜃 < 180°, then find the exact value of sin

𝜃

2.

Complex Numbers

30. Simplify the following complex expression and write your answer in the form 𝑎 + 𝑏𝑖:

4 − 2𝑖

6𝑖

31. Simplify the following complex expression and write your answer in the form 𝑎 + 𝑏𝑖:

4 − 3𝑖

2 + 𝑖

32. Simplify the following complex expression and write your answer in the form 𝑎 + 𝑏𝑖:

7𝑖18 + 3𝑖33 − 8𝑖8 − 5𝑖11

33. Write the simplest possible polynomial with integer coefficients that has a zero of 1 + 2𝑖.

34. Write the simplest possible polynomial with integer coefficients that has zeros of 4 − 𝑖 and √2.

Logarithms and Exponentials

35. Solve: 10003𝑥 = 100𝑥+2

36. Solve, and round your answer to the nearest thousandth: 7𝑥 = 24601

37. Expand the following logarithm as much as possible using the properties of logarithms:

ln (7𝑒3

𝑣4 )

6

38. Expand the following logarithm as much as possible using the properties of logarithms:

log3 (18𝑥2

𝑦3 )

−2

39. Solve for 𝑥: log2 𝑥 + log2(𝑥 − 2) = 3

40. Solve for 𝑥: log5(𝑥2) = 4

41. Evaluate the following logarithm by using the change of base rule: log7 22.

Sequences and Series

42. Find the first term of an arithmetic sequence where 𝑎19 = 72 and 𝑎33 = 114.

43. Find the sum of the first 33 terms of the sequence from the previous problem.

44. Find the first term of a geometric sequence where 𝑎7 = 93750 and 𝑎9 = 2343750.

45. Find the sum of the first eight terms of the geometric sequence where 𝑎3 = 400 and 𝑎6 = −50.

46. In a sequence, 𝑎3 = 10 and 𝑎5 = 40.

a) If the sequence is arithmetic, what is the sum of the first five terms?

b) If the sequence is instead geometric, what is the sum of the first five terms?

Conic Sections

47. Write the equation of a circle with a center of (3, −4) that contains the point (7,0).

48. Write the equation of a circle that has a diameter with endpoints (−4,6) and (12, −2).

49. Graph the conic section on the given coordinate plane. Identify the values of 𝑎, 𝑏, 𝑐, and the

eccentricity.

(𝑥 + 3)2

36−

(𝑦 − 2)2

9= 1

50. Graph the conic section on the given coordinate plane. Identify the values of 𝑎, 𝑏, 𝑐, and the

eccentricity.

(𝑥 − 3)2

25+

(𝑦 + 2)2

36= 1

O

2 4 6 8–2–4–6–8 x

2

4

6

8

–2

–4

–6

–8

y

O

2 4 6 8–2–4–6–8 x

2

4

6

8

–2

–4

–6

–8

y

51. Graph the conic section on the given coordinate plane. Identify the values of 𝑎, 𝑏, 𝑐, and the

eccentricity. [Hint: Start by completing the square.]

−9𝑥2 + 25𝑦2 − 54𝑥 − 200𝑦 + 94 = 0

52. Solve the system of equations: {𝑥2 + 𝑦2 = 10

2𝑥2 − 𝑦2 = 11

53. Solve the system of equations: {𝑥2 + 𝑦2 = 25

𝑦 = −𝑥2 + 5

O

2 4 6 8–2–4–6–8 x

2

4

6

8

–2

–4

–6

–8

y