unit 3: quadratic equations and functions · solving quadratic equations by factoring and graphing...

Honors Algebra 2 ~ Spring 2014 Name_________________ 1

Unit 3: Quadratic Functions and Equations NC Objectives Covered: 1.02 Define and compute with complex numbers 1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems 2.02 Use quadratic functions and inequalities to model and solve problems. a. Solve using graphs. b. Interpret the constants and coefficients in the context of the problem.

Day Date Lesson Assignment

1 Tues.

Feb. 25

Intro to Parabolas/Transformations

Discovery Education Activity(computer lab)

Handout

Packet p. 2

2 Wed.

Feb. 26

Summary of Transformations(notepacket p. 1)

Standard Form to Vertex Form

Vertex Form to Standard Form, Applications

Packet p. 3

3 Thurs.

Feb. 27

Quadratic Regression

“What do you know about Factoring?” Packet p. 4

4 Fri.

Feb. 28

Factoring: GCF, Grouping,

Trinomials, Difference of 2 Squares Packet p. 5

5 Monday

March 3

* Quiz on Sections 5.1-5.4

Solving Quadratic Equations by Factoring

Packet p. 6

6 Tues.

March 4

ACT

Solving Quadratic Equations by Factoring and Graphing

Packet p. 7

7 Wed.

March 5

Review of Radicals

Complex Numbers

“Brochure Project”:

Packet p. 8

EVEN

8 Thurs.

March 6

Complex Numbers

Solve by taking the square root

Packet p. 9

#1-7 odd, #11-18

9 Fri.

March 7

REVIEW

Quiz on days 7-9 Work on Brochure Project

10 Mon.

March 10 Complete the Square/Station Activity Packet p. 10

11 Tues.

March 11

Discriminant

Quadratic Formula Handout

12 Wed.

March 12

Solving Quadratic Inequalities and

Quadratic Systems Packet p. 12 & 13


13 Thurs.

March 13 Review/Quadratic Applications Packet p. 14 & 15

14 Fri.

March 14 Quadratic Functions TEST

Packet p. 17 & 18

Finish Brochure Project

Mon.

March 17 BROCHURE PROJECT DUE!!

Homework Grade: *******************************************************************

Homework Day 1 Part 1: Identify the vertex, the axis of symmetry, the maximum or minimum value, and the domain

and range of each parabola. 2 2 21) 2 1 2) 2 5 3) 3 4 2

Vertex________ Vertex______ Vertex_______

AOS:_________ AOS:_______

y x x y x x y x x

AOS:________

max/min value______ max/min value______ max/min______

domain_________ domain___________ domain_______

range_________ range________ range________

Part 2: Describe how to transform the parent function 2y x to the graph of each function below. 2 2 2

2 2 2

1) 2( 1) 2) 2( 1) 1 3) 3( 2) 3

4) 1( 4) +5 5) 0.25 +3 6) 0.2( 12) - 3

y x y x y x

y x y x y x

Part 3


Homework Day 3

Write in Standard Form. Identify the vertex and y-intercept.

1. y = (x – 3)2 + 1 2. y = -2(x – 3)2 – 6

3. y = -(x + 4)2 – 8 4. y = (x – 8)2 + 3

5. y = -2(x + 7)2 – 10 6. y =2.4(x – 5.1)2 + 3

Determine whether the equations in each pair are equivalent.

7. y = 2(x – 3)2 – 7 8. y = 2(x + 4)2 + 8

y = 2x2 – 12x + 11 y = 2x2 – 16x + 32

Match each equation with the correct statement.

9. y = x2 + 5x + 3 a. The vertex is at (2, 3).

10. y = (x – 2)2 + 3 b. The y-intercept is 5.

11. y = 2(x – 5)2 + 3 c. The y-intercept is 3.

12. y = x2 + 3x + 5 d. The vertex is at (5, 3).

13. The Galleria, in BCE Place in Toronto, has many beautiful parabolic arches. One of the

arches can be modeled by the function y = -0.5(x – 6.8)2 + 26. The x-axis represents the

floor in the Galleria and the y-axis represents the height above the floor. Distances are in

meters.

a. Write the function in standard form.

b. What is the height of the arch at its center?

c. The y-intercept represents the lowest point at one side of the base of an arch. What is

this height?

14. The height, h, of a baseball thrown off a bridge can be modeled by the equation

h = -5(t – 4)2 + 130 where the height is measured in meters and t is the time in seconds since

the ball was thrown.

a. How high was the ball thrown?

b. How long did the ball take to reach its highest point?


1. A toy rocket is shot upward from ground level. The table shows the height of the rocket

at different times.

Time(seconds) 0 1 2 3 4

Height(feet) 0 256 480 672 832

a. Find a quadratic model for this data.

b. Use the model to estimate the height of the rocket after 1.5 seconds.

2. Suppose you are tossing an apple up to a friend on a third-story balcony. After t seconds,

the height of the apple in feet is given by 216 38.4 0.96h t t . Your friend catches the

apple just as it reaches its highest point. How long does the apple take to reach your friend,

and at what height above the ground does your friend catch it?

3. the barber’s profit p each week depends on his charge c per haircut. It is modeled by the

equation 2200 2400 4700p c c . What price should he charge for the largest profit?

4. A skating ring manager finds that revenue R based on an hourly fee F for sakting is

represented by the function 2480 3120R F F . What hourly fee will produce maximum

revenues?

5. The path of a baseball after it has been hit is modeled by the function 20.0032 3h d d , where h is the height in feet of the baseball and d is the distance in

feet the baseball is from home plate. What is the maximum height reached by the baseball?

How far is the baseball from home plate when it reaches its maximum height?

6. A lighting fixture manufacturer has daily production costs of 20.25 10 800C n n ,

Where C is the total cost in dollars and n is the number of light fixtures produced. How

many fixtures should be produced to yield a minimum cost?

7. Find a quadratic model for the data. Use 1981 as year 0.

Price of First-Class Stamp

Year 1981 1991 1995 1999 2001 2006 2007 2008

Price(Cents) 18 29 32 33 34 39 41 42

a. What is the quadratic model?

b. Describe a reasonable domain and range for your model. (Hint: This is a real situation.)

c. Predict when the cost of a stamp will be 50 cents. Is this valid? Why or why not?


Factoring Practice

1. x2 – 18x + 80

2. 5x2y - x2 + 5y - 1 3. 3y2 – 15y + 18

4. a3 – a2b + ab2 – b3

5. x4 – 15x3 + 56x2 6. k2 – 8k + 16

7. 2z2 – 12z + 18

8. c4 + c3 – 12c - 12 9. 25y2 – 100

10. x2 + 3x + xk + 3k

11. a2 – 2a + ad – 2d 12. x2 - 25

REVIEW: Part A: For each function, the vertex of the function’s graph is given. Find a and b.

2 2

2 2

1. 27; (2, 3) 2. 5; ( 1, 4)

3. 8; (2, 4) 4. ; ( 3,2)

y ax bx y ax bx

y ax bx y ax bx

Part B:

1. The equation for the motion of a projectile fired straight up at an initial velocity of

64 ft/s is h = 64t - 16t2, where h is the height in feet and t is the time in seconds. Find

the time the projectile needs to reach its highest point. How high it will go?


Factor each polynomial.

1. 2 3 18x x 2.

25 13 6x x

3. 23 5 2x x 4.

22 1x x

5. 22 3x x 6.

23 11 10x x

7. 2 121x 8.

29 26 3x x

9. 28 18 9x x 10.

22 4 30x x

11. 3 25 20 15x x x 12.

3 24 48 144x x x

13. 216 1x 14.

4 3 218 12 2x x x

15. 4 16x 16.

3 22 6 12x x x

17. 3 23 4 12x x x 18.

23 24x x

19. 4 3 24 24 32x x x 20.

23 6 72x x

21. 416 81x 22.

3 4x x


Solving Quadratic Equations

Solve each equation by factoring.

2 2

2 2

2

1. x - 4x - 12 = 0 2. y - 16y + 64 = 0

3. n + 25 = 10n 4. 9z = 10z

5. 7x = 4x 2

2 2

2 2

2

6. c = 2c + 99

7. 5w - 35w + 60 = 0 8. 3d + 24d + 45 = 0

9. 15v +19v + 6 = 0 10. 4j + 6 = 11j

11. 36k = 25 3 2

3 2 2

12. 12m - 8m = 15m

13. 6e = 5e + 6e 14. 9 = 64p

Solve each equation by graphing. Round each answer to two decimal places.

2 2 2

2 2

15. x 5 3 0 16. 6x 19 15 17. 3x 5 4 0

18. 4x 3 1 19. 10x 3 11

x x x

x x 2 20. 2x 2 5 0 x


Simplifying Expressions Containing Complex Numbers


Simplifying Expressions Containing Complex Numbers Continued

Find the value of m and n for which equation is true.

11) 8 15 2 3 12) ( 2 ) (2 ) 5 5

13) (2 5 ) (1 ) 2 4 14) (4 ) (3 ) 8 2

Solving

i m ni m n m n i i

m n m i i n m n i i

2 2

2 2

using square roots.

15) n 5 4 16) n 8 80

17) 8b 7 193 18) 3 4 85x


Completing the square: Homework day 10

Solve each equation by completing the square. Give the exact answer.

Part 2: Find the value of k that would make the left side of each equation a perfect square

trinomial.

2 2 2

2 2

1. 25 0 2. 100 0 3. 64 0

4. 9 4 0 5. 81 0

x kx x kx x kx

x kx x kx 2 1 6. 0

4x kx

2 2

2 2

2

1. 2 3 0 2. 4v 16 65

3. p 16 22 0 4. n 18 86 0

5. r 2 33 0 6.

a a v

p n

r 2

2 2

2 2

2

a 2 48 0

7. m 12 26 0 8. x 12 20 0

9. 3k 12 6 0 10. p 2 63 0

11. 4m 40 12

a

m x

k p

m 2 12. 7p 28 50 0p


Homework Day 11: Completing the square

Solve each quadratic by completing the square. Give the exact answer. 2 2

2 2

2

1. 2 6 4 2. 12 10

3. 4 5 8 4 4. 2 8 45

5. 2 6 5

x x x x

x x x x

x x 2 6. 3 6 9x x

Part 2: Solve by factoring, completing the square, or taking the square root.

2 2

2 2

2

1. 2 6 6 2. 6 13 6 0

3. ( 3) 9 4. 4 0

5. 2 6 6

x x x x x

x x x

x x x 2 6. 6 9 15 0x x


24 ft

6

Quadratic Applications

1. A smoke jumper jumps from a plane that is 1700 feet above the ground. The function

y = -16x2 + 1700 gives a jumper’s height y in feet after x seconds.

a. How long is the jumper in free fall if the parachute opens at 1000 ft?

b. How long is the jumper in free fall if the parachute opens at 940 ft?

2. You want to expand the garden below by planting a border of flowers. The border will have

the same width around the entire garden. The flowers you bought will fill an area of 276 ft2.

How wide should the border be?

3. One side of a rectangular garden is 2 yd less than the other side. The area of the

garden is 63 yd2. Find the dimensions of the garden.

4. An electronics company has a new line of portable radios with CD players. Their research

suggests that the daily sales s for the new product can be modeled by s = -p2 + 120p + 1400,

where p is the price of each unit.

a. Find the vertex of the function.

b. What is the maximum daily sales total for the new product?

c. What price should the company charge to make this profit?

5. The shape of the Gateway Arch in St. Louis is a catenary curve, which closely resembles a

parabola. The function xxy 4315

2 2 closely models the shape of the arch, where y is the

height in feet and x is the horizontal distance from the base of the left side of the arch in

feet.

a. Graph the function and find its vertex.

b. What is the maximum height of the arch?

c. What is the width of the arch at the base?

16

x

x


Graphing Quadratic Inequalities 2 21. 2 2. 4y x y x

2 23. 8 5 4. 2 12 17y x x y x x

Solving Quadratic Inequalities Solve each inequality 1. x2-x-20 > 0 2. x2-10x+16 < 0 3. x2+4x+3 ≤ 0

4. x2+5x ≥ 24 5. 5x2+10 ≥ 27x 6. 9x2+31x+12 ≤ 0

7. 9z ≤ 12z2 8. 4t2 < 9 9. x2+64 ≥ 16x

10. 4x2+4x+1 > 0


1. A ball is thrown straight up with an initial velocity of 56 feet per second. The height of the ball t

seconds after it is thrown is given by the formula

h(t) = 56t – 16t2.

a. What is the height of the ball after 1 second?_______________________

b. What is its maximum height?____________________________________

c. After how many seconds will it return to the ground? _________________

d. When will the ball be 150 feet above the ground?__________________ 2. An object is thrown upward into the air with an initial velocity of 128 feet per second. The

formula h(t) = 128t – 16t2 gives its height above the ground after t seconds.

a. What is the height after 2 seconds?_____________

b. What is the maximum height reached? ___________

c. For how many seconds will the object be in the air?___________

3. A baseball is projected upward from the top of a 448 foot tall building with an initial velocity of

48 feet per second. The distance s of the baseball from the ground at any time t, in seconds, is

given by the equation s = -16t2 + 48t + 448.

a. Find the time it takes for the baseball to strike the ground. ________

b. What is the baseball’s maximum height?___________

4. A rocket is shot upward such that it’s height in feet, h, is given as h = 62t – 5t2, where t is the

number of seconds since liftoff.

a. Approximate the length of time the rocket is above 120 feet. _________

b. When will the rocket hit the ground?__________________

Use the formula 20( ) 16h t v t t where h(t) is the height of an object in feet, 0v is the object's

initial velocity in feet per second, and t is the time in seconds.

5. An arrow is shot upward with a velocity of 64 feet per second. Ignoring the height of the archer,

how long after the arrow is released does it hit the ground?_____________

6. A tennis ball is hit upward with a velocity of 48 feet per second. Ignoring the height of the

tennis player, how long does it take for the ball to fall to the ground?____________


Review Sheet: Unit 3

1. The following function represents the height, h, of a rocket t seconds after it is launched:

h = - (t – 1.35)2 + 1400. When does the rocket reach its maximum height?

2. Bob wants to fence in his backyard using his house as one side of the fence. He has 350 ft of

fencing available. Find the dimensions of the fence needed to maximize the area.

3. The function h = -16t2 + 150t + 3 represents the height of a ball t seconds after it is

thrown. Could it hit a kite flying 400 feet in the air?

4. A ball is thrown up in the air with an initial velocity of 56 feet per second. The height

of the ball t seconds after it is thrown is given by the formula h(t) = 56t – 16t2. What is

the maximum height of the ball? When does it return to the ground?

5. Solve 2x2 – 7x - 30 > 0

6. Solve 8x2 + 10x – 3 0

7. Solve using any method: 3x2 + x = 2

8. Solve using any method 7x2 – 5x + 1 = 0

9. State the discriminant and nature of roots: x2-3x 2 7

10. State the discriminant and nature of roots: x2-x 4 4

11. Solve: 23 48 0x

12. Simplify: (4 5 ) (3 2 ) (5 6 )i i i

13. Simplify: 4 3

9

i

i

14. Simplify: 120

15. Simplify: (3 5 )(4 6 )i i

16. Simplify: 2(4 2 )i

17. Simplify: 7 3

2

i

i

18. A rectangular garden contains 120 ft2 and has a walk of uniform width surrounding it. If the

entire area, including the walk, is 12 ft x 14 ft., how wide is the sidewalk? Factor Completely:

3 219) 4 3 12a a a 3 620) 8 64y x 221) 12 6m m

222) 2 13 7m m 423) 16y Find the vertex and axis of symmetry: 24) 2 4 7y x x 25) 22 7y x x 26) 2( 3) 4y x 27) Write the equation of a parabola that has a vertex at the origin and passes through the point (3, -6). 28) Write the equation of a parabola that has a vertex at (4, -7), and passes through (3,-9).

29) Describe how the parabola 22( 1) 1y x is shifted/different

from 2y x .


Answers to Review Sheet

1. Max at 1.35 sec.

2. dimensions: 87.5 ft x 175 ft

3. no (graphs do not intersect)

4. Max height: 49 ft., and returns to the ground after 3.5 seconds

5. 5

62

x x or x

6. 3 1

2 4x x

7. 2

13

x and x

8. 5 3

14

ix

9. discriminant = -80, so 2 imaginary roots

10. discriminant = 0, so 1 real rational root

11. 4x i 12. 6-3i

13. 33 31

82

i

14. 2 30i

15. 18 38i 16. 12 16i

17. 3 7

2

i

18. 1 foot

19. 2( 3)( 4)a a

20. 2 2 2 48( 2 )( 2 4 )y x y x y x

21. (3m+2)(4m-3)

22. (2m-1)(m+7)

23. 2( 2)( 2)( 4)y y y

24. V=(2, 3); a of s: x=2

25. V=(1.75, -6.125); a of s: x=1.75

26. V=(3,4); a of s: x=3

27. 22

3y x

28. 22( 4) 7y x

29. more narrow, left 1, up 1


Cumulative Review Units 1-3

_____________1. The attendance at a ball game was 400 people. Student tickets cost $2 and adult tickets

cost $3. $1,050 was collected in ticket sales. Which system models the situation if s is

the number of students and a is the number of a is the number of adults?

A. 2s = 3a B. 3s = 2a C. 2s + 3a = 400 D. s + a = 400

400(s + a) = 1,050 1,050(s + a) = 400 s + a = 1,050 2s + 3a = 1,050

_____________2. What system describes this graph below ? (tick marks are one unit apart)

A.

0

2

x

x

xy

B.

0

2

x

x

xy

C.

0

2

x

x

xy

D.

0

2

x

x

xy

____________3. Which of the following is FALSE?

A. 10

01 is the 2x2 identity matrix. B.

68

34 has no inverse.

C.

000

000

000

001

010

100

100

010

001

D. 00

00 is the 2x2 identity matrix for addition.

_____________4. What is the solution of X + 16

50

21

32 ?

A. 35

22 B.

25

22 C.

35

22 D.

15

22 E.

35

22

_____________5. Which product does not exist?

A. 43

210 B.

01

10

10

01 C. 41

3

2 D.

10

01

124

361

_____________6. . Solve the formula for h. hbbA )(2

121

_____________7. A factory can produce 2 products, x and y, with a profit of P = 8x + 18y -900. The

Production of y can exceed x by no more than 200 units. Also, production is limited by the

Constraint 10002yx . What production levels yield maximum profit?

A. x = 200, y = 400 B. x = 1000, y = 0 C. x = 0, y = 200 D. x = 400, y = 600

____________8. What is the value of 32

17 ?

A. 23 B. 19 C. -19 D. -23

x

y


____________9. What are the values of a,b, and c if 437

6

7

63

c

ab

c

a?

A. a = -2, b = 8, c = 1 B. a = -2, b = 4, c = 1 C. a = 2, b = 8, c = -1 D. a = -2, b = 4, c = -1

___________10. What does z equal in the solution of the system

23

72

83

zyx

zyx

zyx

?

___________11. What is the solution of y in the system 452

264

yx

yx ?

___________12. Write the equation of a line with slope 5

4 and passes through (10, 29) in slope-intercept form.

___________13. For a campaign, a company gave away 5,000 toys to children. Toys x and y cost the company

$1.29 and $0.98, respectively. The company spent $5,613. How many of toy x did they give away?

A. 229 B. 2,000 C. 2,200 D. 2,300

___________14. Two pickup trucks have capacities of ¼ and ½ ton. They made a total of 18 round trips to haul 7 ½ tons of

crushed rock to a job site. Which matrix equation could be used to determine how many round trips each

truck made?

A. 5.7

18

112

1

4

1

y

x B.

18

5.7

112

1

4

1

y

x C.

5.7

18

12

1

14

1

y

x D.

18

5.7

12

1

14

1

y

x

___________15. The Coast Guard flies a rescue out of Elizabeth City to Hatteras in the middle of a Nor’easter. It takes

them one hour with a headwind to fly one hundred miles to get there and thirty minutes to fly back. How fast was the wind

and how fast was the plane in still air?

___________16. What is the equation of the graph of an absolute function that opens down with vertex (4,1) and passes

Through the point (6,0)?

A. 142

1xy B. 14

2

1xy C. 142 xy D. 142 xy

___________17. Solve 20314 x .

A. 3

1112 x B. 2

3

111 x C.

3

1112 xorx D. 2

3

111 xorx

__________18. What is )23(2)41(32 iii written as a complex number in standard form?

A. i63 B. i161 C. i143 D. i21

__________19. Which quadratic function has a vertex of (-4,2) and passes through the point (3,9)?

A. 247

1 2xy B. 24

7

1 2xy C. 24

7

1 2xy D. 24

7

1 2xy

__________20. Solve 433

2 2x

A. 63 B. 63 C. 323 D. 323

unit 3: quadratic equations and functions · solving quadratic equations by factoring and graphing...

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