Algebra II FINAL EXAM REVIEW

Key PowerPoint slides from Ch. 7, 8, 12 & 13 Answer KEY for Final Review Form A

Ch.7

3 Types of Polynomials when solving for x-intercepts:

(quadratic examples given)

Quadratic Function:

Does NOT intersect x-axis…

Intersects x-axis… Intersects x-axis…

therefore cannot factor in terms of Real #’s.

but not at integer or fractional values.

at integer or fractional value(s).

Use quadratic formula to find the complex zeros; conjugate pairs

Use quadratic formula to find the real zeros.

Use the x-intercepts to factor & the 0-product rule to find the zeros. (Or use Quadratic formula)

7.6

7.5 Vocabulary

Note: (a+bi) + (a – bi) => 2a

(real #)

Equations of Circles

Equation

Center (h, k) (h, k)

Radius

Equations of Ellipses

Standard form of equation

Center (h, k)

Stretch a = horizontal scale factor

b = vertical scale factor

Ch. 8 Conic Section Formulas

Ch. 8 Equations of Parabolas

Form of Equation

Direction of Opening vertically oriented; (up/downward) horizontally oriented;

(right, left)

Vertex

(h, k)

(h, k)

Axis of Symmetry

x = h

y = k

Focus

Directrix y = k – f x = h – f

Equations of Hyperbolas

Form of Equation

Direction of Opening horizontally;

(left & right)

vertically;

(up & down)

Center Point (h, k) (h, k)

Vertices (h – a, k) & (h + a, k) (h, k – b) & (h, k + b)

Focus

Asymptotes

12.2 Use the Law of Sines to find the missing parts of each triangle.

1.) Set up the Law of Sines

2.) Solve a proportion that only

has ONE variable.

3.) Solve for first variable.

4.) Repeat until all parts have been

found.

12.3 Law of Cosines - Variations

Law of Sines vs. Cosines

Law of Sines Law of Cosines

S-A-A S-A-S

S-S-A (ambiguous) S-S-S

tail

head or

tip

land

12.5 Intro to Vectors

12.5 Vectors Operations

• Standard ∠ position: an angle where one side is on the (+) x-axis & the other is called the terminal side.

• Terminal side: the side of an angle in standard position NOT on the positive x-axis.

• Reference ∆ : a right ∆ drawn connecting terminal side to the x-axis.

• Reference ∠ : the acute angle between the terminal side & the x-axis.

13.1

13.2

Arc Length & Area of a Sector

90

60 45

30

𝝅

𝟐

𝝅

𝟑 𝝅

𝟒 𝝅

𝟔

1, 0

𝟑

𝟐, 𝟏

𝟐

𝟐

𝟐, 𝟐

𝟐

𝟏

𝟐 , 𝟑

𝟐 0, 1

cosecant: the reciprocal of sine. undefined: sinA = 0 or A = 2𝜋𝑛

secant: the reciprocal of cosine.

undefined: cosA = 0 or A = 𝜋

2 + 𝜋n

cotangent: the reciprocal of tangent. undefined: tanA = 0 or A = 𝜋𝑛

y = csc x

y = sin x

y = csc x

13.4

y = k + b∙sin[ (𝒙−𝒉)

𝒂] or y = k + b∙cos[

(𝒙−𝒉)

𝒂]

Amp. = 𝒃 Max. = 𝒃 + k Min. = - 𝒃 + k Period = 2𝝅 𝒂 Phase shift = h Vertical shift = k

Average Value = 𝒎𝒂𝒙.+𝒎𝒊𝒏.

𝟐

13.4 – 13.5 Transformations:

Identities: cos y = sin x

sin x = sin(𝝅 – x)

sin x = cos(𝝅

𝟐 – x)

sin x = cos (x – 𝝅

𝟐)

y

x

a

b c

cos x = 𝑏

𝑐

sin y = 𝑏

𝑐

sin x = 𝑎

𝑐

cos y = 𝑎

𝑐

x

𝝅 – x

1

sin x

D

C A

B

sin(𝜋 – x)

tan A = 𝑠𝑖𝑛𝐴

𝑐𝑜𝑠𝐴

𝑠𝑖𝑛2𝑥 = (sin 𝑥)2

𝑐𝑜𝑠2𝑥 = (cos 𝑥)2

SOH-CAH-TOA

x

y

r

x

r

y

tan

cos

sin

(x, y) = (cosθ, sinθ)

sinθ = 𝑜𝑝𝑝.

ℎ𝑦𝑝.

cosθ = 𝑎𝑑𝑗.

ℎ𝑦𝑝.

tanθ = 𝑜𝑝𝑝.

𝑎𝑑𝑗.

13.6 +

Algebra II Semester II FINAL Exam Review KEY Form A

Name Period Date

Answer each question and show all work clearly on a separate piece of paper.

Chapters 7–8 • Review 1. Rewrite each equation in the forms requested. Solve for the zeros of each quadratic.

G = General form V = Vertex form F = Factored form

a. y = 2(x – 3)(x – 2) G: y = 2x2 – 10x + 12 V: y = 2(x – 2.5)

2 – 0.5 x = 3, 2

b. y = -2(x – 3.5)

2 + 4.5 G: y = -2x

2 + 14x – 20 F: y = -2(x – 2)(x – 5) x = 2, 5

c. y = x

2 – 8x + 7 V: y = (x – 4)

2 – 9 F: y = (x – 7)(x – 1) x = 1, 7

2. Use C =2 + 5i, D = –3 + i, and E = 4 – 7i to evaluate each expression. Give answers in

the form a + bi. Substitution Complex #

a. C + 2E (2 + 5i) + 2(4 – 7i) 13 – 6i

b. D ÷ E (–3 + i) ÷ (4 – 7i) 39 23

41 41i

c. D2 (–3 + i)

2 48 – 14i

d. CD + E (2 + 5i)( –3 + i) + (4 – 7i) –18 – 15i

3. Write an equation for a cubic function with real coefficients, zeros at –5 and 3i,

and y-intercept 5.

Equation: 3 2 915 5

= + 9y x x x

4. Give the vertex and zeros of each quadratic function, and tell whether the vertex is a

maximum or a minimum.

a. y = 5(x – 7)2

+ 2 Vertex: (7, 2) Minimum Zeros: x = 10

57 ± i

b. y = 11(x – 9)(x – 2) Vertex: (5.5, –134.75) Minimum Zeros: x = 2, 9

c. 21= 4 8

2y x x Vertex: (–4, 0) Maximum Zeros: x = –4

5. Sketch a graph of each function. Include any asymptotes as dashed lines.

a.

2

1=

+ 2y

x b.

2 + 3 + 2=

+ 2

x xy

x c.

1= 2 +

+ 2y

x

a.

b.

c.

6. Write an equation in standard form for each conic section & identify the following:

a.

b.

(h, k) = (4, 0)

a = 3 b = 5

Equation:

(h, k) = (0, -3)

a = 1 b = 2 c =

F1 = (h – c, k) F2 = (h + c, k)

= (- , -3) = ( , -3)

Equation:

Chapters 12-13 • Review Form A

7. Find the value of x. Round answers to the nearest tenth.

a. b.

c.

a. 11.4

b. 18.2

c. 100.3°

8. Port Byron is 90 miles directly north of Port Allen. Ship A leaves Port Allen, moving at a

bearing of 45° and a speed of 19 mi/h. At the same time, ship B leaves Port Byron,

moving at a bearing of 120° and a speed of 22 mi/h. Assume that Port Allen is located at

the origin and that due north is in the positive y-direction.

a. After 4 hours, how far is ship A from Port Allen? 76 miles

b. After 4 hours, how far is ship A from Port Byron? about 66 miles

c. After 10 hours, how far apart are the two ships? About 164 miles

9. Find the exact value of each expression.

a. sin 240 = 3

2 b.

5sec

4

= 2 c. 2

tan3

= 3

d. sin (75) = 2 6

4

e.

1 5tan sin

13

= 5

12 f. cos

–1(cos 290) = 70°

10. Circle P has diameter 12 inches. Find the length of arc AB and the area of the shaded

sector.

s =

in. A =

in2

11. Write an equation for each graph.

a.

b.

Possible answer: y = –1 + cos2x

Possible answer: y = –cos2x + 2