lesson 9 homework 1.4 answer key

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Eureka Math Grade 4 Module 1 Lesson 9 Answer Key

Engage ny eureka math 4th grade module 1 lesson 9 answer key, eureka math grade 4 module 1 lesson 9 problem set answer key.

Eureka Math Grade 4 Module 1 Lesson 9 Exit Ticket Answer Key

Eureka Math Grade 4 Module 1 Lesson 9 Homework Answer Key

Question 2. Round to the nearest ten thousand. a. 88,999 ≈ ______________________ Answer: a. 88,999 ≈ 90,000 b. 85,001 ≈ ______________________ Answer: b. 85,001 ≈ 90,000 c. 789,091 ≈ _____________________ Answer: c. 789,091 ≈ 790,000 d. 905,154 ≈ _____________________ Answer: d. 905,154 ≈ 910,000 e. Explain why two problems have the same answer. Write another number that has the same answer when rounded to the nearest ten thousand. Answer: Problem a, b have the same answer because since its a ten thousand difference. The given two numbers 88,999  ,  85,001 are above the mid points So, they have the same answer. The another number that has the same answer when rounded to the nearest ten thousand is 86,345, 87,123, 88,456, 89,789.

Question 3. Round to the nearest hundred thousand. a. 89,659 ≈ ______________________ Answer: 89,659 ≈  100,000 b. 751,447 ≈ _____________________ Answer: 751,447 ≈ 800,000 c. 617,889 ≈ _____________________ Answer: 617,889 ≈  600,000 d. 817,245 ≈ _____________________ Answer: 817,245 ≈  800,000 e. Explain why two problems have the same answer. Write another number that has the same answer when rounded to the nearest hundred thousand. Answer: Problem b, d have the same answer because since its a hundred thousand difference. The given two numbers 751,447  is above the mid points , 817,245 is below the mid point .So, they have the same answer. The another number that has the same answer when rounded to the nearest hundred thousand is  752,447 , 763,447, 771,447  , 781,447 , 791,447 and 827,245 , 837,245 , 847,245.

Eureka Math Grade 4 Module 1 Answer Key

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How to Use a Math Medic Answer Key

Written by Luke Wilcox published 3 years ago

Answer key might be the wrong term here. Sure, the Math Medic answer keys do provide the correct answers to the questions for a lesson, but they have been carefully designed to do much more than this. They are meant to be the official guide to teaching the lesson, providing specific instructions for what to do and say to make a successful learning experience for your students.

Before we look at the details of the answer key, let's make sure we understand the instructional model first.

Experience First, Formalize Later (EFFL)

A typical Math Medic lesson always has the same four parts: Activity, Debrief Activity, QuickNotes, and Check Your Understanding. Here are the cliff notes:

Activity: Students are in groups of 2 - 4 working collaboratively through the questions in the Activity. The teacher is checking in with groups and using questions, prompts, and cues to get students to refine their communication and understanding. As groups finish the activity, the teacher asks students to go to the whiteboard to write up their answers to the questions.

Debrief Activity: In the whole group setting, the teacher leads a discussion about the student responses to the questions in the activity, often asking students to explain their thinking and reasoning about their answers. The teacher then formalizes the learning by highlighting key concepts and introducing new vocabulary, notation, and formulas.

QuickNotes: The teacher uses direct instruction to summarize the learning from the activity in the QuickNotes box - making direct connections to the learning targets for the lesson.

Check Your Understanding: Students are then asked to apply their learning from the lesson to a new context in the Check Your Understanding (CYU) problem. This can be done individually or in small groups. The CYU is very flexible in it's use, as it can be used as an exit ticket, a homework problem, or a quick review the next day.

How Do I See EFFL in the Answer Key?

You will see EFFL in the answer key like this:

Anything written in blue is something we expect our students to produce. This might not be quite what we expect by the end of the lesson, but provides us with a starting point when we move to formalization.

Anything written in red is an idea added by the teacher - the formalization of the learning that happened during the Activity. Students are expected to add these "notes" to their Activity using a red pen or marker.

What Do Students Write Down For Notes?

By the end of the lesson, students will have written down everything you see on the Math Medic Answer Keys. The most important transition is when students finish the Activity and we move to Debrief Activity. "Students, now is the time for you to put down your pencils and get out your your red Paper Mate flair pens" We give each student a Paper Mate flair pen at the beginning of the school year and tell them they must cherish and protect it with their life. They all think we should be sponsored by Paper Mate (anyone have any leads on this?)

The lessons you see on Math Medic are all of the notes we use with our students. We do not have some secret collection of guided notes.

Do Students Have Access to Answer Keys?

Yes! Any student can create a free Math Medic account to get access to the answer keys. We often send students to the website when they are absent from a lesson or when we don't quite finish the lesson in class. We are comfortable with students having access to these answer keys because we do not think Math Medic lessons should be used as a summative assessment or be used for a grade (unless it's for completion). Our lessons are meant to be the first steps in the formative process of learning new concepts.

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Place value; multidigit addition and subtraction.

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x = 4 3 , x = −4 3 x = 4 3 , x = −4 3

y = 3 3 , y = −3 3 y = 3 3 , y = −3 3

x = 7 , x = −7 x = 7 , x = −7

m = 4 , m = −4 m = 4 , m = −4

c = 2 3 i , c = −2 3 i c = 2 3 i , c = −2 3 i

c = 2 6 i , c = −2 6 i c = 2 6 i , c = −2 6 i

x = 2 10 , x = −2 10 x = 2 10 , x = −2 10

y = 2 7 , y = −2 7 y = 2 7 , y = −2 7

r = 6 5 5 , r = − 6 5 5 r = 6 5 5 , r = − 6 5 5

t = 8 3 3 , t = − 8 3 3 t = 8 3 3 , t = − 8 3 3

a = 3 + 3 2 , a = 3 − 3 2 a = 3 + 3 2 , a = 3 − 3 2

b = −2 + 2 10 , b = −2 − 2 10 b = −2 + 2 10 , b = −2 − 2 10

x = 1 2 + 5 2 x = 1 2 + 5 2 , x = 1 2 − 5 2 x = 1 2 − 5 2

y = − 3 4 + 7 4 , y = − 3 4 − 7 4 y = − 3 4 + 7 4 , y = − 3 4 − 7 4

a = 5 + 2 5 , a = 5 − 2 5 a = 5 + 2 5 , a = 5 − 2 5

b = −3 + 4 2 , b = −3 − 4 2 b = −3 + 4 2 , b = −3 − 4 2

r = − 4 3 + 2 2 i 3 , r = − 4 3 − 2 2 i 3 r = − 4 3 + 2 2 i 3 , r = − 4 3 − 2 2 i 3

t = 4 + 10 i 2 , t = 4 − 10 i 2 t = 4 + 10 i 2 , t = 4 − 10 i 2

m = 7 3 , m = −1 m = 7 3 , m = −1

n = − 3 4 , n = − 7 4 n = − 3 4 , n = − 7 4

ⓐ ( a − 10 ) 2 ( a − 10 ) 2 ⓑ ( b − 5 2 ) 2 ( b − 5 2 ) 2 ⓒ ( p + 1 8 ) 2 ( p + 1 8 ) 2

ⓐ ( b − 2 ) 2 ( b − 2 ) 2 ⓑ ( n + 13 2 ) 2 ( n + 13 2 ) 2 ⓒ ( q − 1 3 ) 2 ( q − 1 3 ) 2

x = −5 , x = −1 x = −5 , x = −1

y = 1 , y = 9 y = 1 , y = 9

y = 5 + 15 i , y = 5 − 15 i y = 5 + 15 i , y = 5 − 15 i

z = −4 + 3 i , z = −4 − 3 i z = −4 + 3 i , z = −4 − 3 i

x = 8 + 4 3 , x = 8 − 4 3 x = 8 + 4 3 , x = 8 − 4 3

y = −4 + 3 3 , y = −4 − 3 3 y = −4 + 3 3 , y = −4 − 3 3

a = −7 , a = 3 a = −7 , a = 3

b = −10 , b = 2 b = −10 , b = 2

p = 5 2 + 61 2 , p = 5 2 − 61 2 p = 5 2 + 61 2 , p = 5 2 − 61 2

q = 7 2 + 37 2 , q = 7 2 − 37 2 q = 7 2 + 37 2 , q = 7 2 − 37 2

c = −9 , c = 3 c = −9 , c = 3

d = 11 , d = −7 d = 11 , d = −7

m = −7 , m = −1 m = −7 , m = −1

n = −2 , n = 8 n = −2 , n = 8

r = − 7 3 , r = 3 r = − 7 3 , r = 3

t = − 5 2 , t = 2 t = − 5 2 , t = 2

x = − 3 8 + 41 8 , x = − 3 8 − 41 8 x = − 3 8 + 41 8 , x = − 3 8 − 41 8

y = 5 3 + 10 3 , y = 5 3 − 10 3 y = 5 3 + 10 3 , y = 5 3 − 10 3

y = 1 , y = 2 3 y = 1 , y = 2 3

z = 1 , z = − 3 2 z = 1 , z = − 3 2

a = −3 , a = 5 a = −3 , a = 5

b = −6 , b = −4 b = −6 , b = −4

m = −6 + 15 3 , m = −6 − 15 3 m = −6 + 15 3 , m = −6 − 15 3

n = −2 + 2 6 5 , n = −2 − 2 6 5 n = −2 + 2 6 5 , n = −2 − 2 6 5

a = 1 4 + 31 4 i , a = 1 4 − 31 4 i a = 1 4 + 31 4 i , a = 1 4 − 31 4 i

b = − 1 5 + 19 5 i , b = − 1 5 − 19 5 i b = − 1 5 + 19 5 i , b = − 1 5 − 19 5 i

x = −1 + 6 , x = −1 − 6 x = −1 + 6 , x = −1 − 6

y = 1 + 2 , y = 1 − 2 y = 1 + 2 , y = 1 − 2

c = 2 + 7 3 , c = 2 − 7 3 c = 2 + 7 3 , c = 2 − 7 3

d = 9 + 33 4 , d = 9 − 33 4 d = 9 + 33 4 , d = 9 − 33 4

r = −5 r = −5

t = 4 5 t = 4 5

ⓐ 2 complex solutions; ⓑ 2 real solutions; ⓒ 1 real solution

ⓐ 2 real solutions; ⓑ 2 complex solutions; ⓒ 1 real solution

ⓐ factoring; ⓑ Square Root Property; ⓒ Quadratic Formula

ⓐ Quadratic Forumula; ⓑ Factoring or Square Root Property ⓒ Square Root Property

x = 2 , x = − 2 , x = 2 , x = −2 x = 2 , x = − 2 , x = 2 , x = −2

x = 7 , x = − 7 , x = 2 , x = −2 x = 7 , x = − 7 , x = 2 , x = −2

x = 3 , x = 1 x = 3 , x = 1

y = −1 , y = 1 y = −1 , y = 1

x = 9 , x = 16 x = 9 , x = 16

x = 4 , x = 16 x = 4 , x = 16

x = −8 , x = 343 x = −8 , x = 343

x = 81 , x = 625 x = 81 , x = 625

x = 4 3 x = 2 x = 4 3 x = 2

x = 2 5 , x = 3 4 x = 2 5 , x = 3 4

The two consecutive odd integers whose product is 99 are 9, 11, and −9, −11

The two consecutive even integers whose product is 128 are 12, 14 and −12, −14.

The height of the triangle is 12 inches and the base is 76 inches.

The height of the triangle is 11 feet and the base is 20 feet.

The length of the garden is approximately 18 feet and the width 11 feet.

The length of the tablecloth is approximatel 11.8 feet and the width 6.8 feet.

The length of the flag pole’s shadow is approximately 6.3 feet and the height of the flag pole is 18.9 feet.

The distance between the opposite corners is approximately 7.2 feet.

The arrow will reach 180 feet on its way up after 3 seconds and again on its way down after approximately 3.8 seconds.

The ball will reach 48 feet on its way up after approximately .6 second and again on its way down after approximately 5.4 seconds.

The speed of the jet stream was 100 mph.

The speed of the jet stream was 50 mph.

Press #1 would take 12 hours, and Press #2 would take 6 hours to do the job alone.

The red hose take 6 hours and the green hose take 3 hours alone.

ⓐ up; ⓑ down

ⓐ down; ⓑ up

ⓐ x = 2 ; x = 2 ; ⓑ ( 2 , −7 ) ( 2 , −7 )

ⓐ x = 1 ; x = 1 ; ⓑ ( 1 , −5 ) ( 1 , −5 )

y -intercept: ( 0 , −8 ) ( 0 , −8 ) x -intercepts ( −4 , 0 ) , ( 2 , 0 ) ( −4 , 0 ) , ( 2 , 0 )

y -intercept: ( 0 , −12 ) ( 0 , −12 ) x -intercepts ( −2 , 0 ) , ( 6 , 0 ) ( −2 , 0 ) , ( 6 , 0 )

y -intercept: ( 0 , 4 ) ( 0 , 4 ) no x -intercept

y -intercept: ( 0 , −5 ) ( 0 , −5 ) x -intercepts ( −1 , 0 ) , ( 5 , 0 ) ( −1 , 0 ) , ( 5 , 0 )

The minimum value of the quadratic function is −4 and it occurs when x = 4.

The maximum value of the quadratic function is 5 and it occurs when x = 2.

It will take 4 seconds for the stone to reach its maximum height of 288 feet.

It will 6.5 seconds for the rocket to reach its maximum height of 676 feet.

ⓑ The graph of g ( x ) = x 2 + 1 g ( x ) = x 2 + 1 is the same as the graph of f ( x ) = x 2 f ( x ) = x 2 but shifted up 1 unit. The graph of h ( x ) = x 2 − 1 h ( x ) = x 2 − 1 is the same as the graph of f ( x ) = x 2 f ( x ) = x 2 but shifted down 1 unit.

ⓑ The graph of h ( x ) = x 2 + 6 h ( x ) = x 2 + 6 is the same as the graph of f ( x ) = x 2 f ( x ) = x 2 but shifted up 6 units. The graph of h ( x ) = x 2 − 6 h ( x ) = x 2 − 6 is the same as the graph of f ( x ) = x 2 f ( x ) = x 2 but shifted down 6 units.

ⓑ The graph of g ( x ) = ( x + 2 ) 2 g ( x ) = ( x + 2 ) 2 is the same as the graph of f ( x ) = x 2 f ( x ) = x 2 but shifted left 2 units. The graph of h ( x ) = ( x − 2 ) 2 h ( x ) = ( x − 2 ) 2 is the same as the graph of f ( x ) = x 2 f ( x ) = x 2 but shift right 2 units.

ⓑ The graph of g ( x ) = ( x + 5 ) 2 g ( x ) = ( x + 5 ) 2 is the same as the graph of f ( x ) = x 2 f ( x ) = x 2 but shifted left 5 units. The graph of h ( x ) = ( x − 5 ) 2 h ( x ) = ( x − 5 ) 2 is the same as the graph of f ( x ) = x 2 f ( x ) = x 2 but shifted right 5 units.

f ( x ) = −4 ( x + 1 ) 2 + 5 f ( x ) = −4 ( x + 1 ) 2 + 5

f ( x ) = 2 ( x − 2 ) 2 − 5 f ( x ) = 2 ( x − 2 ) 2 − 5

ⓐ f ( x ) = 3 ( x − 1 ) 2 + 2 f ( x ) = 3 ( x − 1 ) 2 + 2 ⓑ

ⓐ f ( x ) = −2 ( x − 2 ) 2 + 1 f ( x ) = −2 ( x − 2 ) 2 + 1 ⓑ

f ( x ) = ( x − 3 ) 2 − 4 f ( x ) = ( x − 3 ) 2 − 4

f ( x ) = ( x + 3 ) 2 − 1 f ( x ) = ( x + 3 ) 2 − 1

ⓑ ( −4 , −2 ) ( −4 , −2 )

ⓑ ( − ∞ , 2 ] ∪ [ 6 , ∞ ) ( − ∞ , 2 ] ∪ [ 6 , ∞ )

ⓑ ( −1 , 5 ) ( −1 , 5 )

ⓑ ( − ∞ , 2 ] ∪ [ 8 , ∞ ) ( − ∞ , 2 ] ∪ [ 8 , ∞ )

( − ∞ , −4 ] ∪ [ 2 , ∞ ) ( − ∞ , −4 ] ∪ [ 2 , ∞ )

[ −3 , 5 ] [ −3 , 5 ]

[ −1 − 2 , −1 + 2 ] [ −1 − 2 , −1 + 2 ]

( − ∞ , 4 − 2 ) ∪ ( 4 + 2 , ∞ ) ( − ∞ , 4 − 2 ) ∪ ( 4 + 2 , ∞ )

ⓐ ( − ∞ , ∞ ) ( − ∞ , ∞ ) ⓑ no solution

ⓐ no solution ⓑ ( − ∞ , ∞ ) ( − ∞ , ∞ )

Section 9.1 Exercises

a = ± 7 a = ± 7

r = ± 2 6 r = ± 2 6

u = ± 10 3 u = ± 10 3

m = ± 3 m = ± 3

x = ± 6 x = ± 6

x = ± 5 i x = ± 5 i

x = ± 3 7 i x = ± 3 7 i

x = ± 9 x = ± 9

a = ± 2 5 a = ± 2 5

p = ± 4 7 7 p = ± 4 7 7

y = ± 4 10 5 y = ± 4 10 5

u = 14 , u = −2 u = 14 , u = −2

m = 6 ± 2 5 m = 6 ± 2 5

r = 1 2 ± 3 2 r = 1 2 ± 3 2

y = − 2 3 ± 2 2 9 y = − 2 3 ± 2 2 9

a = 7 ± 5 2 a = 7 ± 5 2

x = −3 ± 2 2 x = −3 ± 2 2

c = − 1 5 ± 3 3 5 i c = − 1 5 ± 3 3 5 i

x = 3 4 ± 7 2 i x = 3 4 ± 7 2 i

m = 2 ± 2 2 m = 2 ± 2 2

x = 3 + 2 3 , x = 3 − 2 3 x = 3 + 2 3 , x = 3 − 2 3

x = − 3 5 , x = 9 5 x = − 3 5 , x = 9 5

x = − 7 6 , x = 11 6 x = − 7 6 , x = 11 6

r = ± 4 r = ± 4

a = 4 ± 2 7 a = 4 ± 2 7

w = 1 , w = 5 3 w = 1 , w = 5 3

a = ± 3 2 a = ± 3 2

p = 1 3 ± 7 3 p = 1 3 ± 7 3

m = ± 2 2 i m = ± 2 2 i

u = 7 ± 6 2 u = 7 ± 6 2

m = 4 ± 2 3 m = 4 ± 2 3

x = −3 , x = −7 x = −3 , x = −7

c = ± 5 6 6 c = ± 5 6 6

x = 6 ± 2 i x = 6 ± 2 i

Answers will vary.

Section 9.2 Exercises

ⓐ ( m − 12 ) 2 ( m − 12 ) 2 ⓑ ( x − 11 2 ) 2 ( x − 11 2 ) 2 ⓒ ( p − 1 6 ) 2 ( p − 1 6 ) 2

ⓐ ( p − 11 ) 2 ( p − 11 ) 2 ⓑ ( y + 5 2 ) 2 ( y + 5 2 ) 2 ⓒ ( m + 1 5 ) 2 ( m + 1 5 ) 2

u = −3 , u = 1 u = −3 , u = 1

x = −1 , x = 21 x = −1 , x = 21

m = −2 ± 2 10 i m = −2 ± 2 10 i

r = −3 ± 2 i r = −3 ± 2 i

a = 5 ± 2 5 a = 5 ± 2 5

x = − 5 2 ± 33 2 x = − 5 2 ± 33 2

u = 1 , u = 13 u = 1 , u = 13

r = −2 , r = 6 r = −2 , r = 6

v = 9 2 ± 89 2 v = 9 2 ± 89 2

x = 5 ± 30 x = 5 ± 30

x = −7 , x = 3 x = −7 , x = 3

m = −11 , m = 1 m = −11 , m = 1

n = 1 ± 14 n = 1 ± 14

c = −2 , c = 3 2 c = −2 , c = 3 2

x = −5 , x = 3 2 x = −5 , x = 3 2

p = − 7 4 ± 161 4 p = − 7 4 ± 161 4

x = 3 10 ± 191 10 i x = 3 10 ± 191 10 i

Section 9.3 Exercises

m = −1 , m = 3 4 m = −1 , m = 3 4

p = 1 2 , p = 3 p = 1 2 , p = 3

p = −4 , p = −3 p = −4 , p = −3

r = −3 , r = 11 r = −3 , r = 11

u = −7 ± 73 6 u = −7 ± 73 6

a = 3 ± 3 2 a = 3 ± 3 2

x = −4 ± 2 5 x = −4 ± 2 5

y = −2 , y = 1 3 y = −2 , y = 1 3

x = − 3 4 ± 15 4 i x = − 3 4 ± 15 4 i

x = 3 8 ± 7 8 i x = 3 8 ± 7 8 i

v = 2 ± 2 13 v = 2 ± 2 13

y = −4 , y = 7 y = −4 , y = 7

b = −2 ± 11 6 b = −2 ± 11 6

c = − 3 4 c = − 3 4

q = − 3 5 q = − 3 5

ⓐ no real solutions no real solutions ⓑ 1 1 ⓒ 2 2

ⓐ 1 1 ⓑ no real solutions no real solutions ⓒ 2 2

ⓐ factor factor ⓑ square root square root ⓒ Quadratic Formula Quadratic Formula

ⓐ Quadratic Formula Quadratic Formula ⓑ square root square root ⓒ factor factor

Section 9.4 Exercises

x = ± 3 , x = ± 2 x = ± 3 , x = ± 2

x = ± 15 , x = ± 2 i x = ± 15 , x = ± 2 i

x = ± 1 , x = ± 6 2 x = ± 1 , x = ± 6 2

x = ± 3 , x = ± 2 2 x = ± 3 , x = ± 2 2

x = −1 , x = 12 x = −1 , x = 12

x = − 5 3 , x = 0 x = − 5 3 , x = 0

x = 0 , x = ± 3 x = 0 , x = ± 3

x = ± 11 2 , x = ± 7 x = ± 11 2 , x = ± 7

x = 25 x = 25

x = 4 x = 4

x = 1 4 x = 1 4

x = 1 25 , x = 9 4 x = 1 25 , x = 9 4

x = −1 , x = −512 x = −1 , x = −512

x = 8 , x = −216 x = 8 , x = −216

x = 27 8 , x = − 64 27 x = 27 8 , x = − 64 27

x = 27 512 , x = 125 x = 27 512 , x = 125

x = 1 , x = 49 x = 1 , x = 49

x = −2 , x = − 3 5 x = −2 , x = − 3 5

x = −2 , x = 4 3 x = −2 , x = 4 3

Section 9.5 Exercises

Two consecutive odd numbers whose product is 255 are 15 and 17, and −15 and −17.

The first and second consecutive odd numbers are 24 and 26, and −26 and −24.

Two consecutive odd numbers whose product is 483 are 21 and 23, and −21 and −23.

The width of the triangle is 5 inches and the height is 18 inches.

The base is 24 feet and the height of the triangle is 10 feet.

The length of the driveway is 15.0 feet and the width is 3.3 feet.

The length of table is 8 feet and the width is 3 feet.

The length of the legs of the right triangle are 3.2 and 9.6 cm.

The length of the diagonal fencing is 7.3 yards.

The ladder will reach 24.5 feet on the side of the house.

The arrow will reach 400 feet on its way up in 2.8 seconds and on the way down in 11 seconds.

The bullet will take 70 seconds to hit the ground.

The speed of the wind was 49 mph.

The speed of the current was 4.3 mph.

The less experienced painter takes 6 hours and the experienced painter takes 3 hours to do the job alone.

Machine #1 takes 3.6 hours and Machine #2 takes 4.6 hours to do the job alone.

Section 9.6 Exercises

ⓐ down ⓑ up

ⓐ x = −4 x = −4 ; ⓑ ( −4 , −17 ) ( −4 , −17 )

ⓐ x = 1 x = 1 ; ⓑ ( 1 , 2 ) ( 1 , 2 )

y -intercept: ( 0 , 6 ) ; ( 0 , 6 ) ; x -intercept ( −1 , 0 ) , ( −6 , 0 ) ( −1 , 0 ) , ( −6 , 0 )

y -intercept: ( 0 , 12 ) ; ( 0 , 12 ) ; x -intercept ( −2 , 0 ) , ( −6 , 0 ) ( −2 , 0 ) , ( −6 , 0 )

y -intercept: ( 0 , −19 ) ; ( 0 , −19 ) ; x -intercept: none

y -intercept: ( 0 , 13 ) ; ( 0 , 13 ) ; x -intercept: none

y -intercept: ( 0 , −16 ) ; ( 0 , −16 ) ; x -intercept ( 5 2 , 0 ) ( 5 2 , 0 )

y -intercept: ( 0 , 9 ) ; ( 0 , 9 ) ; x -intercept ( −3 , 0 ) ( −3 , 0 )

The minimum value is − 9 8 − 9 8 when x = − 1 4 . x = − 1 4 .

The maximum value is 6 when x = 3.

The maximum value is 16 when x = 0.

In 5.3 sec the arrow will reach maximum height of 486 ft.

In 3.4 seconds the ball will reach its maximum height of 185.6 feet.

20 computers will give the maximum of $400 in receipts.

He will be able to sell 35 pairs of boots at the maximum revenue of $1,225.

The length of the side along the river of the corral is 120 feet and the maximum area is 7,200 square feet.

The maximum area of the patio is 800 feet.

Section 9.7 Exercises

ⓑ The graph of g ( x ) = x 2 + 4 g ( x ) = x 2 + 4 is the same as the graph of f ( x ) = x 2 f ( x ) = x 2 but shifted up 4 units. The graph of h ( x ) = x 2 − 4 h ( x ) = x 2 − 4 is the same as the graph of f ( x ) = x 2 f ( x ) = x 2 but shift down 4 units.

ⓑ The graph of g ( x ) = ( x − 3 ) 2 g ( x ) = ( x − 3 ) 2 is the same as the graph of f ( x ) = x 2 f ( x ) = x 2 but shifted right 3 units. The graph of h ( x ) = ( x + 3 ) 2 h ( x ) = ( x + 3 ) 2 is the same as the graph of f ( x ) = x 2 f ( x ) = x 2 but shifted left 3 units.

f ( x ) = −3 ( x + 2 ) 2 + 7 f ( x ) = −3 ( x + 2 ) 2 + 7

f ( x ) = 3 ( x + 1 ) 2 − 4 f ( x ) = 3 ( x + 1 ) 2 − 4

ⓐ f ( x ) = ( x + 3 ) 2 − 4 f ( x ) = ( x + 3 ) 2 − 4 ⓑ

ⓐ f ( x ) = ( x + 2 ) 2 − 1 f ( x ) = ( x + 2 ) 2 − 1 ⓑ

ⓐ f ( x ) = ( x − 3 ) 2 + 6 f ( x ) = ( x − 3 ) 2 + 6 ⓑ

ⓐ f ( x ) = − ( x − 4 ) 2 + 0 f ( x ) = − ( x − 4 ) 2 + 0 ⓑ

ⓐ f ( x ) = − ( x + 2 ) 2 + 6 f ( x ) = − ( x + 2 ) 2 + 6 ⓑ

ⓐ f ( x ) = 5 ( x − 1 ) 2 + 3 f ( x ) = 5 ( x − 1 ) 2 + 3 ⓑ

ⓐ f ( x ) = 2 ( x − 1 ) 2 − 1 f ( x ) = 2 ( x − 1 ) 2 − 1 ⓑ

ⓐ f ( x ) = −2 ( x − 2 ) 2 − 2 f ( x ) = −2 ( x − 2 ) 2 − 2 ⓑ

ⓐ f ( x ) = 2 ( x + 1 ) 2 + 4 f ( x ) = 2 ( x + 1 ) 2 + 4 ⓑ

ⓐ f ( x ) = − ( x − 1 ) 2 − 3 f ( x ) = − ( x − 1 ) 2 − 3 ⓑ

f ( x ) = ( x + 1 ) 2 − 5 f ( x ) = ( x + 1 ) 2 − 5

f ( x ) = 2 ( x − 1 ) 2 − 3 f ( x ) = 2 ( x − 1 ) 2 − 3

Section 9.8 Exercises

ⓑ ( − ∞ , −5 ) ∪ ( −1 , ∞ ) ( − ∞ , −5 ) ∪ ( −1 , ∞ )

ⓑ [ −3 , −1 ] [ −3 , −1 ]

ⓑ ( − ∞ , −6 ] ∪ [ 3 , ∞ ) ( − ∞ , −6 ] ∪ [ 3 , ∞ )

ⓑ [ −3 , 4 ] [ −3 , 4 ]

( − ∞ , −4 ] ∪ [ 1 , ∞ ) ( − ∞ , −4 ] ∪ [ 1 , ∞ )

( 2 , 5 ) ( 2 , 5 )

( − ∞ , −5 ) ∪ ( −3 , ∞ ) ( − ∞ , −5 ) ∪ ( −3 , ∞ )

[ 2 − 2 , 2 + 2 ] [ 2 − 2 , 2 + 2 ]

( − ∞ , 5 − 6 ) ∪ ( 5 + 6 , ∞ ) ( − ∞ , 5 − 6 ) ∪ ( 5 + 6 , ∞ )

( − ∞ , − 5 2 ] ∪ [ − 2 3 , ∞ ) ( − ∞ , − 5 2 ] ∪ [ − 2 3 , ∞ )

[ − 1 2 , 4 ] [ − 1 2 , 4 ]

( − ∞ , ∞ ) . ( − ∞ , ∞ ) .

no solution

Review Exercises

y = ± 12 y = ± 12

a = ± 5 a = ± 5

r = ± 4 2 i r = ± 4 2 i

w = ± 5 3 w = ± 5 3

p = −1 , 9 p = −1 , 9

x = 1 4 ± 3 4 x = 1 4 ± 3 4

n = 4 ± 10 2 n = 4 ± 10 2

n = −5 ± 2 3 n = −5 ± 2 3

( x + 11 ) 2 ( x + 11 ) 2

( a − 3 2 ) 2 ( a − 3 2 ) 2

d = −13 , −1 d = −13 , −1

m = −3 ± 10 i m = −3 ± 10 i

v = 7 ± 3 2 v = 7 ± 3 2

m = −9 , −1 m = −9 , −1

a = 3 2 ± 41 2 a = 3 2 ± 41 2

u = −6 ± 2 2 u = −6 ± 2 2

p = 0 , 6 p = 0 , 6

y = − 1 2 , 2 y = − 1 2 , 2

c = − 1 3 ± 2 7 3 c = − 1 3 ± 2 7 3

x = 3 2 ± 1 2 i x = 3 2 ± 1 2 i

x = 1 4 , 1 x = 1 4 , 1

r = −6 , 7 r = −6 , 7

v = −1 ± 21 8 v = −1 ± 21 8

m = −4 ± 10 3 m = −4 ± 10 3

a = 5 12 ± 23 12 i a = 5 12 ± 23 12 i

u = 5 ± 21 u = 5 ± 21

p = 4 ± 5 5 p = 4 ± 5 5

c = − 1 2 c = − 1 2

ⓐ 1 ⓑ 2 ⓒ 2 ⓓ 2

ⓐ factor ⓑ Quadratic Formula ⓒ square root

x = ± 2 , x = ± 2 3 x = ± 2 , x = ± 2 3

x = ± 1 , x = ± 1 2 x = ± 1 , x = ± 1 2

x = 16 x = 16

x = 64 , x = 216 x = 64 , x = 216

Two consecutive even numbers whose product is 624 are 24 and 26, and −24 and −26.

The height is 14 inches and the width is 10 inches.

The length of the diagonal is 3.6 feet.

The width of the serving table is 4.7 feet and the length is 16.1 feet.

The speed of the wind was 30 mph.

One man takes 3 hours and the other man 6 hours to finish the repair alone.

ⓐ up ⓑ down

x = 2 ; ( 2 , −7 ) x = 2 ; ( 2 , −7 )

y : ( 0 , 15 ) x : ( 3 , 0 ) , ( 5 , 0 ) y : ( 0 , 15 ) x : ( 3 , 0 ) , ( 5 , 0 )

y : ( 0 , −46 ) x : none y : ( 0 , −46 ) x : none

y : ( 0 , −64 ) x : ( −8 , 0 ) y : ( 0 , −64 ) x : ( −8 , 0 )

The maximum value is 2 when x = 2.

The length adjacent to the building is 90 feet giving a maximum area of 4,050 square feet.

f ( x ) = 2 ( x − 1 ) 2 − 6 f ( x ) = 2 ( x − 1 ) 2 − 6

ⓐ f ( x ) = 3 ( x − 1 ) 2 − 4 f ( x ) = 3 ( x − 1 ) 2 − 4 ⓑ

ⓐ f ( x ) = −3 ( x + 2 ) 2 + 7 f ( x ) = −3 ( x + 2 ) 2 + 7 ⓑ

ⓑ ( − ∞ , −2 ) ∪ ( 3 , ∞ ) ( − ∞ , −2 ) ∪ ( 3 , ∞ )

[ −2 , 1 ] [ −2 , 1 ]

( 2 , 4 ) ( 2 , 4 )

[ 3 − 5 , 3 + 5 ] [ 3 − 5 , 3 + 5 ]

Practice Test

w = −2 , w = −8 w = −2 , w = −8

m = 1 , m = 3 2 m = 1 , m = 3 2

y = 2 3 y = 2 3

y = 1 , y = −27 y = 1 , y = −27

ⓐ down ⓑ x = −4 x = −4 ⓒ ( −4 , 0 ) ( −4 , 0 ) ⓓ y : ( 0 , 16 ) ; x : ( −4 , 0 ) y : ( 0 , 16 ) ; x : ( −4 , 0 ) ⓔ minimum value of −4 −4 when x = 0 x = 0

( − ∞ , − 5 2 ) ∪ ( 2 , ∞ ) ( − ∞ , − 5 2 ) ∪ ( 2 , ∞ )

The diagonal is 3.8 units long.

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Access for free at https://openstax.org/books/intermediate-algebra/pages/1-introduction

• Authors: Lynn Marecek
• Publisher/website: OpenStax
• Book title: Intermediate Algebra
• Publication date: Mar 14, 2017
• Location: Houston, Texas
• Book URL: https://openstax.org/books/intermediate-algebra/pages/1-introduction
• Section URL: https://openstax.org/books/intermediate-algebra/pages/chapter-9

© Feb 9, 2022 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

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Lesson 6.1.1, lesson 6.1.2, lesson 6.1.3, lesson 6.1.4, lesson 6.2.1, lesson 6.2.2, lesson 6.2.3, lesson 6.2.4, lesson 6.2.5, lesson 6.2.6, lesson 6.2.7.

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