LESSON Finding 24 x Again?mrpunpanichgulmath.weebly.com/uploads/3/7/5/3/37534823/a1_m4_… · 671...
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671
LESSON
24 LEARNING OBJECTIVES
➤ Today I am: comparing two methods for fi nding the x-intercepts.
➤ So that I can: determine when to use each method.
➤ I’ll know I have it when I can: identify the advantages and disadvantages of
each method.
Finding x-Intercepts Again?
Opening Discussion
The quadratic function, y 5 x2 2 6x 1 8, can be written as y 5 (x 2 2)(x 2 4) and as y 5 (x 2 3)2 2 1. Deshi and Ame wanted to fi nd the x-intercepts of this function. Their work is shown below.
1. Read over each method and then discuss which method you think is the easiest to use.
Deshi’s Method: Using Factored Form Ame’s Method: Using Vertex Form
y 5 (x 2 2)(x 2 4)
0 5 (x 2 2)(x 2 4)
(x 2 2) 5 0 or (x 2 4) 5 0
x 5 2 or x 5 4
y 5 (x 2 3)2 2 1
0 5 (x 2 3)2 2 1
1 5 (x 2 3)2
61 5 x 2 3
x 5 61 1 3
x 5 1 1 3 5 4 or x 5 21 1 3 5 2
y o
rti staff
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672 Module 4 Quadratic Functions
Practice Exercises
For each quadratic function, use the form given to determine the x-intercepts. Your partner will use the other form. Then check that you are getting the same x-intercepts. Be sure to switch methods for each problem.
2. A. Using Factored Form to Find x-Intercepts B. Using Vertex Form to Find x-Intercepts
y 5 (x 1 1)(x 2 3) y 5 (x 2 1)2 2 4
3. A. Using Factored Form to Find x-Intercepts B. Using Vertex Form to Find x-Intercepts
y 5 2(x 2 4)(x 1 2) y 5 2(x 2 1)2 2 18
y o
O cxtlKX 3 O CX 124
4 41 0 X 3 0f 2TI X 3 4 CX l12 X 1I 11
1121 2 3 1 2 1
O 2CX 4 X 12 0 24 15 18118 118
X 4 2 18 221 022T9 cxTI3 X 1113 X
11 4 2
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Unit 10 Completing the Square and the Quadratic Formula Lesson 24 Finding x-Intercepts Again? 673
4. A. Using Factored Form to Find x-Intercepts B. Using Vertex Form to Find x-Intercepts
y x x= − +13
( 7)( 1) y x= − −13
( 3) 163
2
5. A. Using Factored Form to Find x-Intercepts B. Using Vertex Form to Find x-Intercepts
y 5 (2x 2 1)(2x 1 1) y 5 4x2 2 1
O f CX 7 Cx 11 0 15 35 1Z 3 Cx 35
7 I F6 cx1 4 X 3314 X
7 I
0 2 1 2 1 1 I I
2 1 0 2 1 0 I 1 42
54x L x I I
2 2
It _x
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674 Module 4 Quadratic Functions
Discussion
6. Avery prefers the vertex form of quadratic functions. He found the x-intercepts for y 5 (x 2 5)2 2 12 in the following way.
A. Write in each step Avery took.
y 5 (x 2 5)2 2 12
O 5 (x 2 5)2 2 12
12 5 (x 2 5)2
± = −x12 5
= ±x 5 12
= + = −x or x5 2 3 5 2 3
B. What is different about the equation Avery worked with as compared to the ones you used in Exercises 1–5?
7. Kenja really likes the vertex form but she ran into a problem with y 2 10 5 (x 2 1)2. What problem did Kenja face? What are the x-intercepts? What does this mean?
8. Neema wondered how to write the equation of the graph at the right in factored form. What should Neema do to get an equation of this function? Is there a factored form?
Let y oAdd 12square root bothsidesAdd 5Simplify T2
b
Theyshouldbe zero O lo ex D2Flo xThere's no Xintercept
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Unit 10 Completing the Square and the Quadratic Formula Lesson 24 Finding x-Intercepts Again? 675
9. Write an equation that has no x-intercepts. Exchange equations with your partner and prove there are no x-intercepts for your partner’s equation.
10. Marine thought she made a mistake when finding the x-intercepts of the equation y 5 2(x 2 3)(x 2 3) and got only one x-intercept of 3. Explain what Marine’s graph would look like and why this isn’t a mistake.
11. Write the advantages and disadvantages to each method.
Advantages Disadvantages
Usi
ng F
acto
red
Form
to
Fin
d x-
Inte
rcep
tsU
sing
Ver
tex
Form
to
Fin
d x-
Inte
rcep
ts
12. Are there any advantages to using the standard form to get the x-intercepts? Explain your thinking.
Or n
3
FastEasy to find Factorintercepted short 4 long
NosquarerootCI specialcases
Good if the Have to completethe
answer is irrational Square
Fl T2 longerremember toput
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676 Module 4 Quadratic Functions
Practice Exercises—Finding x-Intercepts
For each equation below, determine the x-intercepts, if there are any. You may use any method.
13. y 5 (x 2 5)(3x 1 2) 14. y 5 (4x 1 1)(2x 2 1) 15. y 5 23(x 1 4)(2x 1 3)
16. y 5 2(x 2 3)2 2 50 17. y 5 23(x 1 4)2 1 12 18. y 1 10 5 14
(x 2 1)2
19. y 5 2x2 1 4x 1 4 20. y 5 x2 1 10x 1 9 21. y 2 11 5 x2 2 6x
O X 5 3 2
X 5 0X 5
3 12 0
39 2
23
0 3 x145 12 0 10 14 X 112 3CX14123 3
T T 10 14CxD2
4 Cx 1412 x 14 IT4o cx4 4 44412 X Ianto X lG 2 II
112To X12510,1 air
0 2 2 4 4completethesquarecannotfactor
0 2622 1 2foggy
4
0 2 X1172 22 22 2 X115FT cxNo real x intercepts
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Unit 10 Completing the Square and the Quadratic Formula Lesson 24 Finding x-Intercepts Again? 677
Practice Exercises—Finding the Vertex
For each equation below, determine the vertex. You may use any method.
22. y 5 (x 2 5)(3x 1 2) 23. y 5 (4x 1 1)(2x 2 1) 24. y 5 23(x 1 4)(2x 1 3)
25. y 5 2(x 2 3)2 2 50 26. y 5 23(x 1 4)2 1 12 27. y 1 10 5 14
(x 2 1)2
28. y 5 2x2 1 4x 1 4 29. y 5 x2 1 10x 1 9 30. y 2 11 5 x2 2 6x
Lesson Summary
When a quadratic equation is not conducive to factoring, we can solve by completing
the square.
Completing the square can be used to find solutions that are irrational, something
very difficult to do by factoring.
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O XI GX 12
O x 761.9
12 9
O x 1332 21
IT CX 3
IDT XI3
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Unit 10 Completing the Square and the Quadratic Formula Lesson 24 Finding x-Intercepts Again? 679
NAME: PERIOD: DATE:
Homework Problem Set
1. Solve the equation for b: 2b2 2 9b 5 3b2 2 4b 2 14.
2. Solve for x. 12 5 x2 1 6x
3. Solve for x. 4x2 2 40x 1 93 5 0
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680 Module 4 Quadratic Functions
Solve each equation by completing the square.
4. x2 2 2x 5 12
5. 12
r2 2 6r 5 2 (Hint: Consider multiplying every term by 2.)
6. 2p2 1 8p 5 26
7. Challenge 2y2 1 3y 2 5 5 4
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Unit 10 Completing the Square and the Quadratic Formula Lesson 24 Finding x-Intercepts Again? 681
Solve each equation. Use any method.
8. p2 2 2p 5 8
9. 2q2 1 8q 5 4
10. 13
m2 1 2m 1 8 5 5
11. 24x2 5 24x 1 11
12. Challenge Rewrite the expression by completing the square: 12
b2 2 4b 1 13.
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682 Module 4 Quadratic Functions
Determine the x-intercepts of each quadratic function, if there are any.
13. y 5 (2x 2 1)(x 1 2) 14. y 5 x(4x 1 1) 15. y 5 (x 2 7)(2x 2 5)
16. y 5 (x 2 4)2 2 1 17. y 5 2(x 1 3)2 2 2 18. y 1 16 5 (x 2 2)2
19. y 5 3x2 2 7x 20. y 5 x2 2 5x 2 24 21. y 5 25x2 2 1