unit 3: quadratic equations and...
TRANSCRIPT
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
Honors Algebra 2 ~ Spring 2014 Name_________________ 2
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
Honors Algebra 2 ~ Spring 2014 Name_________________ 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?
Honors Algebra 2 ~ Spring 2014 Name_________________ 4
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?
Honors Algebra 2 ~ Spring 2014 Name_________________ 5
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?
Honors Algebra 2 ~ Spring 2014 Name_________________ 6
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
Honors Algebra 2 ~ Spring 2014 Name_________________ 7
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
Honors Algebra 2 ~ Spring 2014 Name_________________ 8
Simplifying Expressions Containing Complex Numbers
Honors Algebra 2 ~ Spring 2014 Name_________________ 9
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
Honors Algebra 2 ~ Spring 2014 Name_________________ 10
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
Honors Algebra 2 ~ Spring 2014 Name_________________ 11
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
Honors Algebra 2 ~ Spring 2014 Name_________________ 12
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
Honors Algebra 2 ~ Spring 2014 Name_________________ 13
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
Honors Algebra 2 ~ Spring 2014 Name_________________ 14
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?____________
Honors Algebra 2 ~ Spring 2014 Name_________________ 15
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 .
Honors Algebra 2 ~ Spring 2014 Name_________________ 16
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
Honors Algebra 2 ~ Spring 2014 Name_________________ 17
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
Honors Algebra 2 ~ Spring 2014 Name_________________ 18
____________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