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4.7 Review

4.7 Review

Assessment

Presentation

•

Mathematics

•

10th - 11th Grade

•

Easy

Created by

Jim Cross

Used 18+ times

FREE Resource

13 Slides • 34 Questions

1

​Unit 4

Rational Expressions and Equations
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2

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​When multilying rational expressions, factor everything you can, cancel out common factors in numerator and denominator and then multiply across

3

To find restricted values:


make any denominator with a variable in it equal to zero and solve. This is easier when the denominators are factored

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4

Multiple Choice

Complete Step 1:  FACTOR x2 −5x + 6x2−4⋅(x +6)(x + 5)x2+2x −15\frac{x^2\ -5x\ +\ 6}{x^2-4}\cdot\frac{\left(x\ +6\right)\left(x\ +\ 5\right)}{x^2+2x\ -15}  

1

(x − 3)(x − 2)(x + 2)(x − 2)⋅(x + 6)(x + 5)(x + 5)(x − 3)\frac{\left(x\ -\ 3\right)\left(x\ -\ 2\right)}{\left(x\ +\ 2\right)\left(x\ -\ 2\right)}\cdot\frac{\left(x\ +\ 6\right)\left(x\ +\ 5\right)}{\left(x\ +\ 5\right)\left(x\ -\ 3\right)}  

2

(x + 6)(x − 1)(x + 2)(x − 2)⋅(x + 6)(x + 5)(x −5)(x −3)\frac{\left(x\ +\ 6\right)\left(x\ -\ 1\right)}{\left(x\ +\ 2\right)\left(x\ -\ 2\right)}\cdot\frac{\left(x\ +\ 6\right)\left(x\ +\ 5\right)}{\left(x\ -5\right)\left(x\ -3\right)}  

5

Multiple Choice

Complete Step 2: State the Restrictions (x−3)(x−2)(x+2)(x−2)⋅(x+6)(x+5)(x+5)(x−3)\frac{\left(x-3\right)\left(x-2\right)}{\left(x+2\right)\left(x-2\right)}\cdot\frac{\left(x+6\right)\left(x+5\right)}{\left(x+5\right)\left(x-3\right)}  

1

x≠−2, 2, −5, 3x\ne-2,\ 2,\ -5,\ 3  

2

x≠2, −2, 5, −3x\ne2,\ -2,\ 5,\ -3  

3

x≠3, 2, −6, −5x\ne3,\ 2,\ -6,\ -5  

4

x≠−3, 2, −6, −5x\ne-3,\ 2,\ -6,\ -5  

6

Multiple Choice

Complete Step 3:  CANCEL (x−3)(x−2)(x+2)(x−2)⋅(x+6)(x+5)(x+5)(x−3)\frac{\left(x-3\right)\left(x-2\right)}{\left(x+2\right)\left(x-2\right)}\cdot\frac{\left(x+6\right)\left(x+5\right)}{\left(x+5\right)\left(x-3\right)}  

1

x+6x+2\frac{x+6}{x+2}  

2

x+2x+6\frac{x+2}{x+6}  

3

(x−3)(x+2)⋅(x+6)(x−3)\frac{\left(x-3\right)}{\left(x+2\right)}\cdot\frac{\left(x+6\right)}{\left(x-3\right)}  

7

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​When dividng rational expressions, switch it to multiplication times the reciprocal of the second fraction.

Then treat it like a multiplication problem

8

restricted values for divsion or rational expressions

Remember that since you are going to take the reciprocal of the fraction you divide by, both its numerator and denominator will need to be checked for restricted values.

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9

Multiple Choice

Complete Step 1: we do not divide by frations, instead we chooswe 3(x−9)x2−36÷x−9x2−x−30\frac{3\left(x-9\right)}{x^2-36}\div\frac{x-9}{x^2-x-30}  

1

x2−363(x−9)⋅x2−x−30x−9\frac{x^2-36}{3\left(x-9\right)}\cdot\frac{x^2-x-30}{x-9}  

2

3(x−9)x2−36⋅x2−x−30x−9\frac{3\left(x-9\right)}{x^2-36}\cdot\frac{x^2-x-30}{x-9}  

10

Multiple Choice

Complete Step 2:  FACTOR 3(x−9)x2−36⋅x2−x−30x−9\frac{3\left(x-9\right)}{x^2-36}\cdot\frac{x^2-x-30}{x-9}  

1

3(x−9)(x+6)(x+6)⋅(x−5)(x−6)x−9\frac{3\left(x-9\right)}{\left(x+6\right)\left(x+6\right)}\cdot\frac{\left(x-5\right)\left(x-6\right)}{x-9}  

2

3(x−9)(x+6)(x−6)⋅(x−6)(x +5)x−9\frac{3\left(x-9\right)}{\left(x+6\right)\left(x-6\right)}\cdot\frac{\left(x-6\right)\left(x\ +5\right)}{x-9}  

11

Multiple Choice

Complete Step 3:  State Restrictions (recall this was originally a division question)

3(x−9)(x+6)(x−6)⋅(x−6)(x+5)x−9\frac{3\left(x-9\right)}{\left(x+6\right)\left(x-6\right)}\cdot\frac{\left(x-6\right)\left(x+5\right)}{x-9}  

1

x≠−6, 6, 9, −5x\ne-6,\ 6,\ 9,\ -5  

2

3(x−9)(x+6)(x−6)⋅(x−6)(x +5)x−9\frac{3\left(x-9\right)}{\left(x+6\right)\left(x-6\right)}\cdot\frac{\left(x-6\right)\left(x\ +5\right)}{x-9}  

3

x≠9, −6, −5x\ne9,\ -6,\ -5  

4

x≠−9, −6, 5x\ne-9,\ -6,\ 5  

12

Multiple Choice

Complete Step 4:  CANCEL 3(x−9)(x+6)(x−6)⋅(x−6)(x+5)x−9\frac{3\left(x-9\right)}{\left(x+6\right)\left(x-6\right)}\cdot\frac{\left(x-6\right)\left(x+5\right)}{x-9}  

1

3(x+5)x+6\frac{3\left(x+5\right)}{x+6}  

2

x+5x+6\frac{x+5}{x+6}  

3

3(x+6)(x+5)\frac{3}{\left(x+6\right)\left(x+5\right)}  

13

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14

Multiple Choice

Subtract 6x−1−5x4\frac{6}{x-1}-\frac{5x}{4}  

1

11x4(x−1)\frac{11x}{4\left(x-1\right)}  

2

1x4(x−1)\frac{1x}{4\left(x-1\right)}  

3

6−5x4(x−1)\frac{6-5x}{4\left(x-1\right)}  

4

−5x2+5x+244(x−1)\frac{-5x^2+5x+24}{4\left(x-1\right)}  

15

Multiple Choice

Add 2x−3+3x+2\frac{2}{x-3}+\frac{3}{x+2}  

1

5x−5(x−3)(x+2)\frac{5x-5}{\left(x-3\right)\left(x+2\right)}  

2

2x−1(x−3)(x+2)\frac{2x-1}{\left(x-3\right)\left(x+2\right)}  

3

5x−1\frac{5}{x-1}  

4

1−2x(x−3)(x+2)\frac{1-2x}{\left(x-3\right)\left(x+2\right)}  

16

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17

Multiple Select

Solve      3cc2−4+1c−2=2c+2\frac{3c}{c^2-4}+\frac{1}{c-2}=\frac{2}{c+2}  

1

-3

2

3

3

2

4

no solution

18

Multiple Select

Solve for x.

4x2−8x+12=xx−2+1x−6\frac{4}{x^2-8x+12}=\frac{x}{x-2}+\frac{1}{x-6}  

1

-1

2

2

3

6

4

No solution

19

Multiple Choice

Question image

Solve for x:
hint: when it is a

fraction = fraction

cross multiply

1
x = 0 , 4
2
x = 1 , 3
3
x = -4 , 0
4
x = 5 , 2

20

Multiple Choice

Question image
Solve
1
20 min
2
40 min
3
1 hour
4
2 hour

21

Multiple Choice

After a heavy snowfall, Brian can shovel all of the driveway in 30 minutes. If his younger brother Allen helps, the job takes only 20 minutes. How long would it take Allen to do the job by himself?
1
60 minutes
2
45 minutes
3
50 minutes
4
555 minutes

22

4.7 Review
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23

Finding Horizontal Asymptotes

There are 3 situations you need to worry about.

1. degree is bigger on the bottom,

HA is y = 0

​

2. degrees are the same,

divide leading coefficients

​

3. degree is bigger on top,

there is no HA

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24

Multiple Choice

Question image

Find the horizontal asymptote.

1

None

2

y=-2

3

y=2

4

y=0

25

Multiple Choice

Question image

Find the horizontal asymptote.

1

y=−2y=-2

2

none

3

y=0y=0

4

y=12y=\frac{1}{2}

26

Multiple Choice

Question image
1

y = 0

2

y = 1/2

3

y = 2

4

None

27

Multiple Choice

Question image

Find the horizontal asymptote.

1

None

2

y=0y=0  

3

y=−13y=\frac{-1}{3}  

4

y=−3y=-3  

28

Points of Discontinuity: Holes and VA

1. factor both numerator and denominator


2. any factors that cancel out that could cause the denominator to equal zero will create a hole


3. any remaining factors in the denominator that can cause the denominator to = 0 will create vertical asymptotes.

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29

Multiple Choice

What is an asymptote?

1

an imaginary line that your function never touches

2

a part of your function

3

a point on your graph

4

a type of fruit

30

Multiple Choice

What causes a hole in a graph

1

any squared factors

2

factor in denominator that cancels out

3

factor in numerator = 0

4

a factor that cancels out = 0

31

Multiple Choice

What causes a vertical asymptote in a rational graph

1

any squared factors

2

factor = 0 in denominator after holes have been identified

3

factor in numerator = 0

4

a factor that cancels out = 0

32

Multiple Choice

Question image

What is the Vertical Asymptotes?

1

x= -5

2

x= 5

3

x= 6

4

x= -6

33

Multiple Choice

Question image

What is the hole?

1

x= 4

2

x= -4

3

x= 5

4

x= -5

34

Multiple Choice

Question image

What are the Vertical Asymptotes of...

1

x = 0, -5

2

x=0, 5

3

x = 2, -7

4

x = -2, 7

35

Multiple Choice

Question image

What are the domain?

1

x > 5

2

x≠5x\ne5

3

x < 5

36

Multiple Choice

Question image

What is the domain?

1

x≠1x\ne1

2

x≠1, 3x\ne1,\ 3  

3

x≠−1,  −3x\ne-1,\ \ -3  

4

x≠3x\ne3

37

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​Points of discontinuity are holes and vertical aysmptotes.

Factor the denominator and try to answer the question.

38

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​Points of discontinuity are holes and vertical aysmptotes.

Factor the denominator and try to answer the question.


the denominator will be

(x + 6)(x - 1)

hole will be at x = 1
VA will be at x = -6

pts of discontinuity are
x = 1, x = -6

39

Multiple Choice

Identify the transformations & Asymptotes for f(x)=−1x+3f\left(x\right)=-\frac{1}{x+3}  

1

Move Right 3 and Up 0

VA: x = 3

HA: y = 0

2

Move Left 3 and Up 0

VA: x = -3

HA: y = 0

3

Move Right 3 and Down 1

VA: x = 3

HA: y = -1

4

Move Left 3 and Down 1

VA: x = -3

HA: y = -1

40

Multiple Choice

Question image

What are the points of discontinuity?

(holes or vertical asymptotes)

1

x = {5, -5}

2

x = { -5, 4}

3

x= {1, 5, 4}

4

x = {5, -4}

41

Multiple Choice

Question image

What are the points of discontinuity?

(Holes or vertical asymptotes)

1

x = { 5, 2}

2

x = {5, 0}

3

x = {-5, 5}

4

x = {-5, -2}

42

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43

Multiple Choice

Referring to f(x)=  1x\frac{1}{x}  how does f(x) =  1x+3\frac{1}{x+3}  translate?. 

1

Shifts right 3

2

Shifts up three

3

Shifts left three

4

Shifts down three

44

Multiple Choice

Referring to f(x) =  1x\frac{1}{x}  how does f(x) =   1x−4+1\frac{1}{x-4}+1 Translate. 

1

Shifts left 4 units and up 1 unit.

2

Shifts left 4 units and down 1 unit.

3

Shifts right 4 units and up 1 unit.

4

Shifts right 4 units and down 1 unit.

45

Multiple Choice

Identify the transformations & Asymptotes for f(x)=3x−2+2f\left(x\right)=\frac{3}{x-2}+2  

1

Move Right 2 and Up 2

VA: x = 2

HA: y = 2

2

Move Left 2 and Up 3

VA: x = -2

HA: y = 3

3

Move Right 2 and Down 2

VA: x = 2

HA: y = -2

4

Move Left 2 and Down 3

VA: x = -2

HA: y = -3

46

Multiple Choice

Question image
What is the vertical asymptote? 
1
x = 2
2
x = 7
3
y = 2
4
y = 7

47

Multiple Choice

Question image
What is the horizontal asymptote? 
1
x = 2
2
x = 7
3
y = 2
4
y = 7
pattern-tertiary
​Unit 4

Rational Expressions and Equations
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