Modeling Data With Quadratic Functions

1
Modeling Data With Quadratic Functions
ALGEBRA 2 LESSON 5-1
(For help, go to Lessons 1-2 and 2-2.)
1. ƒ(x) x 2. ƒ(x) x2 3. ƒ(x)
x 4. ƒ(x) x2 5. ƒ(x) x2 6. ƒ(x)
x2 7. 3x 4y 8 8. 2x y
7 9. x 3y 9
Evaluate each function for x 3, 1, 0, 1, and
3.
1 3
1 3
Write each equation in slope-intercept form.
1 2
5-1
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Modeling Data With Quadratic Functions
ALGEBRA 2 LESSON 5-1
Solutions
1. ƒ(3) 3 ƒ(1) 1 ƒ(0) 0 ƒ(1) 1
ƒ(3) 3 2. ƒ(3) (3)2 (3)(3) 9 ƒ(1)
(1)2 (1)(1) 1 ƒ(0) 02 0 0 0
ƒ(1) 12 1 1 1 ƒ(3) 32 3 3
9 3. ƒ(3) (3) 3 ƒ(1) (1) 1 ƒ(0)
0 0 ƒ(1) 1 ƒ(3) 3 4. ƒ(3) (3)2
(3)(3) 9 ƒ(1) (1)2 (1)(1)
1 ƒ(0) 02 (0 0) 0 0 ƒ(1) 12
(1 1) 1 ƒ(3) 32 (3 3) 9
5-1
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Modeling Data With Quadratic Functions
ALGEBRA 2 LESSON 5-1
Solutions (continued)
5. ƒ(3) 3 ƒ(1) ƒ(0) 0 ƒ(1)
ƒ(3) 3 6. ƒ(3) 3 ƒ(1) ƒ(0) 0
ƒ(1) ƒ(3) 3 7. 3x 4y 8 8. 2x
y 7 4y 3x 8 y 2x
7 4y 3x 8 y 2x 7
4 4 y x 2
9. y x 3
1 3
1 3
1 3
1 3
3 4
1 6
5-1
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Modeling Data With Quadratic Functions
ALGEBRA 2 LESSON 5-1
Determine whether each function is linear or
quadratic. Identify the quadratic, linear, and
constant terms.
a. ƒ(x) (2x 1)2
This is a quadratic function.
Quadratic term 4x2 Linear term 4x Constant
term 1
5-1
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Modeling Data With Quadratic Functions
ALGEBRA 2 LESSON 5-1
(continued)
b. ƒ(x) x2 (x 1)(x 1)
This is a linear function.
Quadratic term none Linear term 0x (or
0) Constant term 1
5-1
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Modeling Data With Quadratic Functions
ALGEBRA 2 LESSON 5-1
Below is the graph of y x2 6x 11. Identify
the vertex and the axis of symmetry. Identify
points corresponding to P and Q.
5-1
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Modeling Data With Quadratic Functions
ALGEBRA 2 LESSON 5-1
Find the quadratic function to model the values
in the table.
The solution is a 2, b 7, c 5. Substitute
these values into standard form. The quadratic
function is y 2x2 7x 5.
5-1
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Modeling Data With Quadratic Functions
ALGEBRA 2 LESSON 5-1
The table shows data about the wavelength x (in
meters) and the wave speed y (in meters per
second) of deep water ocean waves. Use the
graphing calculator to model the data with a
quadratic function. Graph the data and the
function. Use the model to estimate the wave
speed of a deep water wave that has a wavelength
of 6 meters.
Wavelength (m) 3 5 7 8
Wave Speed (m/s) 6 16 31 40
5-1
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Modeling Data With Quadratic Functions
ALGEBRA 2 LESSON 5-1
(continued)
An approximate model of the quadratic function is
y 0.59x2 0.34x 0.33.
At a wavelength of 6 meters the wave speed is
approximately 23m/s.
5-1
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Modeling Data With Quadratic Functions
ALGEBRA 2 LESSON 5-1
Pages 237239 Exercises
1. linear none, x, 4 2. quadratic 2x2, 3x,
5 3. quadratic 3x2, 6x, none 4. quadratic
x2, none, 7 5. quadratic x2, 3x,
10 6. linear none, 7x, 28 7. quadratic
6x2, none, 6 8. linear none, x, 1
9. quadratic 2x2, 8x, none 10. (0, 4), x
0 11. (1, 0), x 1 12. (1, 4), x
1 13. P'(6, 9), Q'(2, 1) 14. P'(1, 5), Q'(2,
8) 15. P'(1, 1), Q'(4, 4) 16. y x2 3x
4
17. y x2 5x 2 18. y 2x2 x 3 19. y
x2 2x 20. y 3x2 20 21. a. y 16x2
33x 46, where x is the number of seconds
after release and y is height in
feet. b. 28.5 ft
5-1
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Modeling Data With Quadratic Functions
ALGEBRA 2 LESSON 5-1
1 2
1 2
22. a. y 0.0236x2 0.907x
2.09 b. 58.5 23. y 4x2 24. y 2x2
3x 5 25. no 26. y x2 x 1 27.
, , x 28. (1, 4), x 1
29. ,0 , x 30. a. x 4, 5 y 6,
10 b. y x2 x c. 45 segments 31. a. y
0.01467x2 1.312x 9.795 b. 1992
31. (continued) c. Never the quadratic model
is useful over a limited number of years, but
because it increases and then decreases, it
does not model the data after
2021. 32. 3 33. 8 34. 6
1 2
1 2
5 8
7 4
1 2
1 2
1 2
5-1
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Modeling Data With Quadratic Functions
ALGEBRA 2 LESSON 5-1
11 8
35. 36. 37. 10 38. a. y 3.157x
52.34 b. y 0.04243x2 0.04080x 0.8890
38. (continued) c. Answers may vary. Sample
Quadratic the quadratic model comes closer to
most data points than the linear model because
the data follows a curve. 39. Answers may
vary. Sample y x2, y x2  
x, y x2 x 40. Answers may vary. Sample
You need at least 3 points you are going to
substitute x- and y-values into y ax2 bx c
to set up and solve a linear system for finding
values of a, b, and c.
25 4
2 5
1 5
6 5
5-1
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Modeling Data With Quadratic Functions
ALGEBRA 2 LESSON 5-1
41. Answers may vary. Sample They are similar in
that both are symmetric with respect in the
y-axis, have only non-negative y-values, lie in
Quadrants I and II, and have minimums at (0, 0)
they are different in that the graph of y x2
rises more steeply, while y x rises at a
steady rate as x increases. 42. (3,
5) 43. a. You can find how high the arrow was
when it was released. b. The negative
intercept tells you how much earlier you would
have to shoot from a height of zero the arrow
for its height to be described by the same
function. The positive intercept tells you how
many seconds after the release the arrow will
take to hit the ground.
44. A 45. H 46. A 47. I
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Modeling Data With Quadratic Functions
ALGEBRA 2 LESSON 5-1
50. 51. 0   4   17   4 52. 53. (2,
5) 54. (5, 8) 55. (1, 1)
48. 4 System a(1)2 b(1) c 6, a(2)2
b(2) c 11, a(3)2 b(3) c 20 a 2, b
1, c 5 y 2x2 x 5 3 appropriate
methods, but with one computational
error 2 incorrect system solved correctly
OR correct system solved incorrectly 1 cor
rect function, without work shown 49.
(3, 2)
5-1
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Modeling Data With Quadratic Functions
ALGEBRA 2 LESSON 5-1
4 5
56. 57. 58. 1
1 2
5-1
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Modeling Data With Quadratic Functions
ALGEBRA 2 LESSON 5-1
1. Below is the graph of y x2 2x 2.
Identify the vertex and axis of symmetry.
Identify points corresponding to P and
Q. Find a quadratic function to model the
values in each table. 2. 3.
y 3x2 x 4
y 2x2 5x 6
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Properties of Parabolas
ALGEBRA 2 LESSON 5-2
(For help, go to Lessons 2-2 and 2-5.)
Find the y-intercept of the graph of each
function.
1. y 3x 3 2. y 2x 1 3. 4x 3y
12 Find the vertex of the graph of each
function. 4. y 2x 5. y x
1 6. y 3x 7 Graph each equation. 7. y
4x 3 8. x y 2 9. y 5x 5
2 3
1 2
5-2
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Properties of Parabolas
ALGEBRA 2 LESSON 5-2
Solutions
1. y 3x 3 y-intercept 3 2. y 2x 1
y-intercept 1 3. 4x 3y 12 4. y 2x
0, 3y 4x 12
3y 4x 12 3 3
y x 4 y-intercept 4 5. y
x 1 6. y 3x 7
, vertex ( , 0) vertex ( ,
0)
b m
0 2
vertex (0, 0)
4 3
(1)
2 3
b m
7 3
3 2
b m
2 3
3 2
7 3
5-2
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Properties of Parabolas
ALGEBRA 2 LESSON 5-2
Solutions (continued)
7. y 4x 3 slope 4 y-intercept
3 9. y 5x 5
1 vertex (1, 0) y-intercept point (0,
5), its reflection (2, 5)
b m
5 5
5-2
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Properties of Parabolas
ALGEBRA 2 LESSON 5-2
1 3
Graph y x2 1.
5-2
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Properties of Parabolas
ALGEBRA 2 LESSON 5-2
1 2
Graph y x2 x 3. Name the vertex and
axis of symmetry.
5-2
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Properties of Parabolas
ALGEBRA 2 LESSON 5-2
(continued)
Step 4 Evaluate the function for another value
of x, such as y (2)2 (2) 3 7. Graph
(2, 7) and its reflection (4, 7).
1 2
5-2
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Properties of Parabolas
ALGEBRA 2 LESSON 5-2
1 4
Graph y x2 2x 3. What is the maximum
value of the function?
Since a lt 0, the graph of the function opens
down, and the vertex represents the maximum
value. Find the coordinates of the vertex.
5-2
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Properties of Parabolas
ALGEBRA 2 LESSON 5-2
The number of weekend get-away packages a hotel
can sell is modeled by 0.12p 60, where p is
the price of a get-away package. What price will
maximize revenue? What is the maximum revenue?
5-2
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Properties of Parabolas
ALGEBRA 2 LESSON 5-2
(continued)
Find the maximum value of the function. Since a lt
0, the graph of the function opens down, and the
vertex represents a maximum value.
R 0.12(250)2 60(250) Evaluate R for p 250
7500 Simplify.
A price of 250 will maximize revenue. The
maximum revenue is 7500.
5-2
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Properties of Parabolas
ALGEBRA 2 LESSON 5-2
Pages 244247 Exercises
1. 2.
3. 4.
5. 6.
5-2
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Properties of Parabolas
ALGEBRA 2 LESSON 5-2
7. 8.
9. 10. 11.
12. 13.
5-2
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Properties of Parabolas
ALGEBRA 2 LESSON 5-2
14. 15.
16. 17.
18. 19. 20.
5-2
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Properties of Parabolas
ALGEBRA 2 LESSON 5-2
1 4
10 3
21. 22. max, 6
23. min, 24. max, 25.
min, 2
26. max, 6 27. min,
5 28. 10 13,500 29. 2 s 64 ft
41 8
5-2
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Properties of Parabolas
ALGEBRA 2 LESSON 5-2
30. 1000 tires 20 31. 32.
33. 34.
35. 13, 13 169 36. 5, 5 25 37. B 38. C
39. A 40. Answers may vary. Sample y x2
20x 96 41. 2.25 ft by 2.25 ft 5.0625 ft2
5-2
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Properties of Parabolas
ALGEBRA 2 LESSON 5-2
5-2
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Properties of Parabolas
ALGEBRA 2 LESSON 5-2
5-2
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Properties of Parabolas
ALGEBRA 2 LESSON 5-2
5-2
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Properties of Parabolas
ALGEBRA 2 LESSON 5-2
83. Let x amount of storage space and y
amount of space to be covered by the roof. x
40,000 y 25,000
5-2
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Properties of Parabolas
ALGEBRA 2 LESSON 5-2
1. Graph y 2x2 4x 2. Label the vertex and
axis of symmetry.
2. What is the minimum value of the function y
3x2 2x  8?
3. The equation h 40t 16t2 describes the
height h, in feet, of a ball that is thrown
straight up as a function of the time t, in
seconds, that the ball has been in the air. At
what time does the ball reach its maximum height?
What is the maximum height?
1.25 s 25 ft
5-2
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Translating Parabolas
ALGEBRA 2 LESSON 5-3
(For help, go to Lesson2-6.)
Identify the parent function of each function.
Then graph the function by translating the parent
function.
1. y x 2 2. y 3x 2 3. y x 1
1 4. y 2x, 2 units down 5. y x, 4 units
up, 1 unit right
Write an equation for each translation.
5-3
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Translating Parabolas
ALGEBRA 2 LESSON 5-3
Solutions
1. y x 2 parent function y
x translate 2 units up 3. y x 1
1 parent function y x translate 2 unit left
and one unit down
2. y 3x 2 parent function y
3x translate 2 units up
4. y 2x, 2 units down y 2x 2 5. y x, 4
units up and 1 unit right y (x 1) 4, or y
x 3
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Translating Parabolas
ALGEBRA 2 LESSON 5-3
2 3
Graph y (x 1)2 2.
You can graph it by translating the parent
function or by finding the vertex and the axis of
symmetry.
5-3
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Translating Parabolas
ALGEBRA 2 LESSON 5-3
Write the equation of the parabola shown below.
y a(x h)2 k Use the vertex form.
y a(x 2)2 5 Substitute h 2 and k 5.
3 a(0 2)2 5 Substitute (0, 3).
2 4a Simplify.
5-3
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Translating Parabolas
ALGEBRA 2 LESSON 5-3
A long strip of colored paper is attached as a
party decoration at exactly opposite corners of
the back wall of a rectangular party room. The
strip approximates a parabola with equation y
0.008(x 25)2 10. The bottom left corner of
the back wall is the origin and x and y are
measured in feet. How far apart are the side
walls? How high are they?
The function is in vertex form. Since h 25 and
k 10, the vertex is at (25, 10).
The vertex is halfway between the two corners of
the wall, so the width of the wall is 2(25 ft)
50 ft.
5-3
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Translating Parabolas
ALGEBRA 2 LESSON 5-3
(continued)
To find the walls height, find y for x 0.
The wall is 50 ft long and 15 ft high.
5-3
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Translating Parabolas
ALGEBRA 2 LESSON 5-3
Write y 7x2 70x 169 in vertex form.
The vertex is at (5, 6).
The vertex form of the function is y 7(x 5)2
6.
5-3
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Translating Parabolas
ALGEBRA 2 LESSON 5-3
5-3
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Translating Parabolas
ALGEBRA 2 LESSON 5-3
1 4
5-3
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Translating Parabolas
ALGEBRA 2 LESSON 5-3
5-3
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Translating Parabolas
ALGEBRA 2 LESSON 5-3
1 2
5-3
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Translating Parabolas
ALGEBRA 2 LESSON 5-3
3 2
1 2
35 2
1 2
5-3
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Translating Parabolas
ALGEBRA 2 LESSON 5-3
5-3
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Translating Parabolas
ALGEBRA 2 LESSON 5-3
1 5
1 4
5-3
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Translating Parabolas
ALGEBRA 2 LESSON 5-3
5-3
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Translating Parabolas
ALGEBRA 2 LESSON 5-3
1. Graph the function y 4(x 3)2.
2. Identify the vertex and the y-intercept of the
graph of y 2(x 5)2 8.
(5, 8), 42
3. Write the equation y 3x2 12x 1 in vertex
form.
y 3(x 2)2 13
5-3
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Factoring Quadratic Expressions
ALGEBRA 2 LESSON 5-4
(For help, go to Lessons 1-2 and 5-1.)
Simplify each expression.
1. x2 x 4x 1 2. 6x2 4(3)x 2x
3 3. 4x2 2(5 x)  3x 4. 2x(5
x) 5. (2x 7)(2x 7) 6. (4x 3)(4x 3)
Multiply.
5-4
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Factoring Quadratic Expressions
ALGEBRA 2 LESSON 5-4
1. x2 x 4x 1 x2 (1 4)x 1 x2
5x 1 2. 6x2 4(3)x 2x 3 6x2 12x 2x
3 6x2 (12 2)x 3 6x2 10x
3 3. 4x2 2(5 x) 3x 4x2 10 2x 3x
4x2 (2 3)x 10 4x2 x 10 4. 2x(5 x)
2x(5) 2x(x) 10x 2x2 2x2 10x 5. (2x
7)(2x 7) (2x)2 2(2x)(7) (7)2 4x2
28x 49 6. (4x 3)(4x 3) (4x)2 (3)2
16x2 9
Solutions
5-4
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Factoring Quadratic Expressions
ALGEBRA 2 LESSON 5-4
Factor each expression.
a. 15x2 25x 100
15x2 25x 100 5(3x2) 5(5x) 5(20) Factor
out the GCF, 5
5(3x2 5x 20) Rewrite using the
Distributive Property.
b. 8m2 4m
8m2 4m 4m(2m) 4m(1) Factor out the GCF, 4m
4m(2m 1) Rewrite using the Distributive
Property.
5-4
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Factoring Quadratic Expressions
ALGEBRA 2 LESSON 5-4
Factor x2 10x 24.
Step 1 Find factors with product ac and sum b.
Step 2 Rewrite the term bx using the factors you
found. Group the remaining terms and find the
common factors for each group.
5-4
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Factoring Quadratic Expressions
ALGEBRA 2 LESSON 5-4
(continued)
Step 3 Rewrite the expression as a product of
two binominals.
5-4
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Factoring Quadratic Expressions
ALGEBRA 2 LESSON 5-4
Factor x2  14x 33.
Step 1 Find factors with product ac and sum b.
Step 2 Rewrite the term bx using the factors you
found. Then find common factors and rewrite the
expression as a product of two binomials.
x(x 3) 11(x 3) Find common factors.
(x 11)(x 3) Rewrite using the Distributive
Property.
5-4
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Factoring Quadratic Expressions
ALGEBRA 2 LESSON 5-4
Factor x2 3x 28.
Step 1 Find factors with product ac and sum b.
Step 2 Since a 1, you can write binomials
using the factors you found.
x2 3x 28
(x 4)(x 7) Use the factors you found.
5-4
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Factoring Quadratic Expressions
ALGEBRA 2 LESSON 5-4
Factor 6x2 31x 35.
Step 1 Find factors with product ac and sum b.
Step 2 Rewrite the term bx using the factors you
found. Then find common factors and rewrite the
expression as the product of two binomials.
2x(3x 5) 7(3x 5) Find common factors.
(2x 7)(3x 5) Rewrite using the Distributive
Property.
5-4
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Factoring Quadratic Expressions
ALGEBRA 2 LESSON 5-4
Factor 6x2 11x 35.
Step 1 Find factors with product ac and sum b.
Step 2 Rewrite the term bx using the factors you
found. Then find common factors and rewrite the
expression as the product of two binomials.
2x(3x 5) 7(3x 5) Find common factors.
(2x 7)(3x 5) Rewrite using the Distributive
Property.
5-4
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Factoring Quadratic Expressions
ALGEBRA 2 LESSON 5-4
Factor 100x2 180x 81.
100x2 180x 81 (10x)2 180 (9)2 Rewrite
the first and third terms as squares.
(10x)2 180 (9)2 Rewrite the middle term to
verify the perfect square trinomial pattern.
(10x 9)2 a2 2ab b2 (a b)2
5-4
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Factoring Quadratic Expressions
ALGEBRA 2 LESSON 5-4
A square photo is enclosed in a square frame, as
shown in the diagram. Express the area of the
frame (the shaded area) in completely factored
form.
Define Let x length of side of frame.
The area of the frame in factored form is (x
7)(x 7) in2.
5-4
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Factoring Quadratic Expressions
ALGEBRA 2 LESSON 5-4
5-4
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Factoring Quadratic Expressions
ALGEBRA 2 LESSON 5-4
5-4
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Factoring Quadratic Expressions
ALGEBRA 2 LESSON 5-4
5-4
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Factoring Quadratic Expressions
ALGEBRA 2 LESSON 5-4
5-4
67
Factoring Quadratic Expressions
ALGEBRA 2 LESSON 5-4
5-4
68
Factoring Quadratic Expressions
ALGEBRA 2 LESSON 5-4
5-4
69
Factoring Quadratic Expressions
ALGEBRA 2 LESSON 5-4
1. 12x2 6x 18 2. m2 11m 18 3. x2
14x 15 4. x2 13x 42 5. 64x2 144x
81 6. 3x2 5x 50 7. 5k2 125 8. 15n2 8n
1
Factor each expression completely.
6(2x2 x 3)
(m 2)(m 9)
(x 15)(x 1)
(x 6)(x 7)
(8x 9)2
(3x 10)(x 5)
5(k 5)(k 5)
(5n 1)(3n 1)
5-4
70
Quadratic Equations
ALGEBRA 2 LESSON 5-5
(For help, go to Lessons 5-2 and 5-4.)
Factor each expression.
1. x2 5x 14 2. 4x2 12 3. 9x2 16 4. y
x2 2x 5 5. y x2 4x 4 6. y x2 4x
Graph each function.
5-5
71
Quadratic Equations
ALGEBRA 2 LESSON 5-5
(For help, go to Lessons 8-3 and 7-4.)
1. Factors of 14 with a sum of 5 7 and 2 x2
5x 14 (x 7)(x 2) 2. 4x2 12x 4x(x
3) 3. 9x2 16 (3x)2 (4)2 (3x 4)(3x
4) 4. y x2 2x 5 axis of symmetry x
1 vertex (1, 6)
Solutions
(2) 2(1)
5-5
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Quadratic Equations
ALGEBRA 2 LESSON 5-5
(For help, go to Lessons 8-3 and 7-4.)
Solutions (continued)
5. y x2 4x 4 axis of symmetry x
2 vertex (2, 0)
6. y x2 4x axis of symmetry x
2 vertex (2, 4)
(4) 2(1)
5-5
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Quadratic Equations
ALGEBRA 2 LESSON 5-5
3x2 20x  7 0
3x(x 7) (x 7) 0 Find common factors.
(3x 1)(x 7) 0 Factor using the
Distributive Property.
5-5
74
Quadratic Equations
ALGEBRA 2 LESSON 5-5
(continued)
Check 3x2 20x 7 3x2 20x 7
1 3
1 3
3 2 20 7 3(7)2 20(7) 7
1 3
20 3
7 147 140 7
7 7 7 7
5-5
75
Quadratic Equations
ALGEBRA 2 LESSON 5-5
Solve 6x2 486 0.
6x2 486 0 Rewrite in the form ax2 c.
x2 81 Simplify.
x 9 Take the square root of each side.
5-5
76
Quadratic Equations
ALGEBRA 2 LESSON 5-5
The function y 16x2 270 models the height y
in feet of a heavy object x seconds after it is
dropped from the top of a building that is 270
feet tall. How long does it take the object to
hit the ground?
y 16x2 270
0 16x2 270 Substitute 0 for y.
The object takes about 4.1 seconds to hit the
ground.
Check Is the answer reasonable? The negative
number 4.1 is also a solution to the equation.
However, since a negative value for time has no
meaning in this case, only the positive solution
is reasonable.
5-5
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Quadratic Equations
ALGEBRA 2 LESSON 5-5
Use a graphing calculator to solve 2x2 7x 1
0. Round the solution to the nearest hundredth.
5-5
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Quadratic Equations
ALGEBRA 2 LESSON 5-5
A carpenter wants to cut a piece of plywood in
the shape of a right triangle. He wants the
hypotenuse of the triangle to be 6 feet long, as
shown in the diagram. About how long should the
perpendicular sides be?
Relate From the Pythagorean Theorem we know for
a right triangle that the hypotenuse squared
equals the sum of the squares of the other two
sides.
5-5
79
Quadratic Equations
ALGEBRA 2 LESSON 5-5
(continued)
Graph the related function y 2x2 2x 35. Use
the CALC feature to find the positive solution.
The sides of the triangle are 3.7 ft and 4.7 ft.
5-5
80
Quadratic Equations
ALGEBRA 2 LESSON 5-5
5-5
81
Quadratic Equations
ALGEBRA 2 LESSON 5-5
5-5
82
Quadratic Equations
ALGEBRA 2 LESSON 5-5
5-5
83
Quadratic Equations
ALGEBRA 2 LESSON 5-5
5-5
84
Quadratic Equations
ALGEBRA 2 LESSON 5-5
5-5
85
Quadratic Equations
ALGEBRA 2 LESSON 5-5
1. 4x2 17x 15 0 2. 10x2 19x 6
0 3. Solve 3x2 4800 by using square
roots. 4. Use a graphing calculator to solve
2x2 5x 9 0. Round the solutions to the
nearest hundredth.
Solve each equation by factoring.
40
3.71, 1.21
5-5
86
Complex Numbers
ALGEBRA 2 LESSON 5-6
(For help, go to Skills Handbook pg 855.)
Factor each expression.
1. 32 42 2. (2)2 82 3. 52 (12)2 4. 62
102 5. x2 x2 6. (3x)2 (4x)2
5-6
87
Complex Numbers
ALGEBRA 2 LESSON 5-6
1. 32 42 9 16 25 5 2. (2)2
82 4 64 68 4 17 2
17 3. 52 (12)2 25 144 169
13 4. 62 102 36 100 136 4
34 2 34 5. x2 x2 2x2 2
x2 2 x x 2 6. (3x)2 (4x)2
9x2 16x2 25x2 5x
Solutions
5-6
88
Complex Numbers
ALGEBRA 2 LESSON 5-6
Simplify 54 by using the imaginary number i.
5-6
89
Complex Numbers
ALGEBRA 2 LESSON 5-6
Write 121 7 in a bi form.
7 11i Write in the form a bi.
5-6
90
Complex Numbers
ALGEBRA 2 LESSON 5-6
Find each absolute value.
a. 7i
7i is seven units from the origin on the
imaginary axis.
So 7i 7
b. 10 24i
5-6
91
Complex Numbers
ALGEBRA 2 LESSON 5-6
Find the additive inverse of 7 9i.
7 9i
(7 9i) Find the opposite.
7 9i Simplify.
5-6
92
Complex Numbers
ALGEBRA 2 LESSON 5-6
Simplify (3 6i) (4 8i).
(3 6i) (4 8i) 3 (4) 6i 8i Use
commutative and associative properties.
1 14i Simplify.
5-6
93
Complex Numbers
ALGEBRA 2 LESSON 5-6
Find each product.
a. (3i)(8i)
(3i)(8i) 24i 2 Multiply the real numbers.
24(1) Substitute 1 for i 2.
24 Multiply.
b. (3 7i )(2 4i )
(3 7i )(2 4i ) 6 14i 12i 28i 2
Multiply the binomials.
6 26i 28(1) Substitute 1 for i 2.
22 26i Simplify.
5-6
94
Complex Numbers
ALGEBRA 2 LESSON 5-6
Solve 9x2 54 0.
9x2 54 0
9x2 54 Isolate x2.
x2 6
5-6
95
Complex Numbers
ALGEBRA 2 LESSON 5-6
Find the first three output values for f(z) z2
4i. Use z 0 as the first input value.
f(16 4i ) (16 4i )2 4i Second output
becomes third input. Evaluate for z 16 4i.
The first three output values are 4i, 16 4i,
240 124i.
5-6
96
Complex Numbers
ALGEBRA 2 LESSON 5-6
5-6
97
Complex Numbers
ALGEBRA 2 LESSON 5-6
5-6
98
Complex Numbers
ALGEBRA 2 LESSON 5-6
5-6
99
Complex Numbers
ALGEBRA 2 LESSON 5-6
5-6
100
Complex Numbers
ALGEBRA 2 LESSON 5-6
5-6
101
Complex Numbers
ALGEBRA 2 LESSON 5-6
5-6
102
Complex Numbers
ALGEBRA 2 LESSON 5-6
5-6
103
Complex Numbers
ALGEBRA 2 LESSON 5-6
1. Simplify 169. 2. Find 1 i
Simplify each expression. 3. (7i )(3i
) 4. (9 10i) (7 4i) 5. (3 4i ) (3
8i ) 6. (5 2i)(7 4i) 7. Solve x2 2
0.
13i
21
2 14i
12i
27 34i
1 2
2i
5-6
104
Completing the Square
ALGEBRA 2 LESSON 5-7
(For help, go to Lessons 5-1 and page 262.)
5-7
105
Completing the Square
ALGEBRA 2 LESSON 5-7
Solutions
5-7
106
Completing the Square
ALGEBRA 2 LESSON 5-7
Solve x2 12x 36 9.
x2 12x 36 9
(x 6)2 9 Factor the trinomial.
x 6 3 Find the square root of each side.
x 6 3 or x 6 3 Solve for x.
5-7
107
Completing the Square
ALGEBRA 2 LESSON 5-7
Find the missing value to complete the square
x2 20x .
x2 20x 100 Complete the square.
5-7
108
Completing the Square
ALGEBRA 2 LESSON 5-7
Solve x2 4x 1 0.
x2 4x 1 Rewrite so all terms containing x
are on one side.
x2 4x 4 1 4 Complete the square by
adding 4 to each side.
(x 2)2 3 Factor the perfect square trinomal.
5-7
109
Completing the Square
ALGEBRA 2 LESSON 5-7
(continued)
Check
5-7
110
Completing the Square
ALGEBRA 2 LESSON 5-7
Solve x2 6x 12 0.
5-7
111
Completing the Square
ALGEBRA 2 LESSON 5-7
Solve 2x2 7x 1 0.
5-7
112
Completing the Square
ALGEBRA 2 LESSON 5-7
Write y x2 5x 2 in vertex form.
y x2 5x 2
5-7
113
Completing the Square
ALGEBRA 2 LESSON 5-7
The monthly profit P from the sales of rugs
woven by a family rug-making business depends on
the price r that they charge for a rug. The
profit is model by P r 2 500r 59,500.
Write the function in vertex form. Use the vertex
form to find the price that yields the maximum
monthly profit and the amount of the maximum
profit.
P r 2 500r 59500
The vertex is (250, 3000). A price of 250 per
rug gives a maximum monthly profit of 3000.
5-7
114
Completing the Square
ALGEBRA 2 LESSON 5-7
5-7
115
Completing the Square
ALGEBRA 2 LESSON 5-7
5-7
116
Completing the Square
ALGEBRA 2 LESSON 5-7
5-7
117
Completing the Square
ALGEBRA 2 LESSON 5-7
49. row 2 47, 46, 45, 44, 43, 42, 41, 40 row 3
96, 141, 184, 225, 264, 301, 336, 369,
400 a. A w2 50w b. Check
students work. c. The numbers w such that 0 lt w
lt 50 since the perimeter is 100 the width would
have to be less than 50, and since length cant
be negative it would have to be greater than
0.
49. (continued) d. 625 units2 25 units by 25
units e. A w(50 w) yes both equations
are quadratic and model the same situation.
5-7
118
Completing the Square
ALGEBRA 2 LESSON 5-7
5-7
119
Completing the Square
ALGEBRA 2 LESSON 5-7
5-7
120
Completing the Square
ALGEBRA 2 LESSON 5-7
5-7
121
Completing the Square
ALGEBRA 2 LESSON 5-7
5-7
122
Completing the Square
ALGEBRA 2 LESSON 5-7
1. x2 60x 2. x2 7x Simplify each
expression. 3. x2 6x 16 0 4. x2 14x 74
0 5. 3x2 5x  28 0 6. 4x2 6x 3 0
Complete the square.
49 9
900
2, 8
7 5i
5-7
123
The Quadratic Formula
ALGEBRA 2 LESSON 5-8
(For help, go to Lessons 1-2 and 5-1.)
1. y 8 10x2 2. y (x 2)2 1 3. y
2x(x 1) (x 1)2 4. y (3x)2 (x
1)2 5. a 1, b 6, c 3 6. a 5, b 2,
c 4 7. a 3, b 6, c 7 8. a 2, b 3,
c 10
Write each quadratic equation in standard form.
Evaluate the expression b2 4ac for the given
values of a, b, and c.
5-8
124
The Quadratic Formula
ALGEBRA 2 LESSON 5-8
1. y 8 10x2 y 10x2 8 2. y (x 2)2
1 x2 2(2)x 22 1 x2 4x 4 1 x2
4x 3 3. y 2x(x 1) (x 1)2 2x2
2x x2 2x 1 x2 4x 1 4. y (3x)2
(x 1)2 9x2 (x2 2x 1) 8x2 2x
1 5. b2 4ac for a 1, b 6, c 3 62
4(1)(3) 36 12 24 6. b2 4ac for a 5, b
2, c 4 22 4(5)(4) 4 (80) 84 7. b2
4ac for a 3, b 6, c 7 (6)2 4(3)(7)
36 84 48 8. b2 4ac for a 2, b 3, c
10 32 4(2)(10) 9 (80) 89
Solutions
5-8
125
The Quadratic Formula
ALGEBRA 2 LESSON 5-8
Use the Quadratic Formula to solve 3x2 23x
40 0.
5-8
126
The Quadratic Formula
ALGEBRA 2 LESSON 5-8
(continued)
5-8
127
The Quadratic Formula
ALGEBRA 2 LESSON 5-8
Solve 3x2 2x 4.
5-8
128
The Quadratic Formula
ALGEBRA 2 LESSON 5-8
The longer leg of a right triangle is 1 unit
longer than the shorter leg. The hypotenuse is 3
units long. What is the length of the shorter leg?
5-8
129
The Quadratic Formula
ALGEBRA 2 LESSON 5-8
(continued)
5-8
130
The Quadratic Formula
ALGEBRA 2 LESSON 5-8
Determine the type and number of solutions of
x2 5x 10 0.
a 1, b 5, c 10 Find the values of a, b,
and c.
b2  4ac (5)2 4(1)(10) Evaluate the
discriminant.
25 40 Simplify.
15
Since the discriminant is negative, x2 5x 10
0 has two imaginary solutions.
5-8
131
The Quadratic Formula
ALGEBRA 2 LESSON 5-8
A player throws a ball up and toward a wall that
is 17 ft high. The height h in feet of the ball t
seconds after it leaves the players hand is
modeled by h 16t 2 25t 6. If the ball
makes it to where the wall is, will it go over
the wall or hit the wall?
Since the discriminant is negative, the equation
17 16t2 25t 6 has no real solution. The
ball will hit the wall.
5-8
132
The Quadratic Formula
ALGEBRA 2 LESSON 5-8
5-8
133
The Quadratic Formula
ALGEBRA 2 LESSON 5-8
5-8
134
The Quadratic Formula
ALGEBRA 2 LESSON 5-8
5-8
135
The Quadratic Formula
ALGEBRA 2 LESSON 5-8
57. two 58. one 59. none 60. two 61. two
62. two 63. a. 12 or 12 b. k such that k lt
12 c. k such that k gt 12
64. a. k such that k lt 6 b. k such that k gt
6 c. 6, 6 65. Imaginary solutions always come
in pairs because they are the positive and
negative solution of the square root of a
negative number. 66. a. II b. III c. I 67. a
. x2 100 b. 17.72 cm
5-8
136
The Quadratic Formula
ALGEBRA 2 LESSON 5-8
5-8
137
The Quadratic Formula
ALGEBRA 2 LESSON 5-8
5-8
138
The Quadratic Formula
ALGEBRA 2 LESSON 5-8
5-8
139
The Quadratic Formula
ALGEBRA 2 LESSON 5-8
1. 6x2 19x 8 0 2. x2 2x 11
0 3. x2 2x 15 0 4. 3x2 7x 5
0 5. 2x2 5x  7 0 6. 3x2 14x 8
0 7. 4x2 5x 10 7x 1
Use the Quadratic Formula to solve each equation.
3, 5
Find the value of the discriminant for each
quadratic equation. Tell how many different
solutions each equation has and whether the
solutions are real or imaginary.
31 two, imaginary
100 two, real
0 one, real
5-8
140
Quadratic Equations and Functions
ALGEBRA 2 CHAPTER 5
5-A
141
Quadratic Equations and Functions
ALGEBRA 2 CHAPTER 5
5-A
142
Quadratic Equations and Functions
ALGEBRA 2 CHAPTER 5
26. 9, 1 27. 28. 0, 29. ,
30. , 31.
,
21. 2 13 22. 13
23. Answers may vary. Sample In the coordinate
plane you graph ordered pairs (a, b) in the
complex plane you graph complex numbers a bi.
For both, you find a on the horizontal axis and
you find b on the vertical axis. 24. 5,
5 25. 8, 3
,
3 2
1 2
1 2
1 10

1 3
22 2
1 3
5-A
143
Quadratic Equations and Functions
ALGEBRA 2 CHAPTER 5
5-A