collecting/combining like terms - pbworks

MBF3C U2L1 Expanding and Simplifying 2

Topic : Expanding and Simplifying

Goal : I know how to expand a polynomial expression with the distributive law and I can collect like terms in order to simplify the expression.

Collecting/Combining Like TermsSimplify the following

What does simplify mean?

put together anything that is the same (aka collect like terms

any terms that share the exact same variables - this means the variables must have the same exponents.

What are like terms?

How do I combine like terms?

identify the like terms and then combine into one simply by adding the number that is with it.

* pay attention to the sign of that number and obey integer rules

* if there is no number in front we understand it to be one

Example 1. Simplify the following

a)

b)

MBF3C U2L1 Expanding and Simplifying 2

Using the Distributive Law

BEDMAS The Distributive Law

3(5+2) 3(5+2)

The distributive law gives us a way to get around order of operations when brackets are involved. This is good when there are variables in the question because we can't add terms that aren't like.

Example 2. Expand and Simplify (when necessary)

a)

b)

Practice Questions - Handout Page

c)

www.mathworksheetsgo.com

II. Practice

Simplify

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

IV. Challenge Problems

Kelly

Kelly

MBF3C U2L1ws Combining Like Terms


21. Write an expression for the perimeter of the figure below.

22. Write an expression for the perimeter of the rectangle below.

23. Write an expression for the perimeter of the regular hexagon below.

24. Write the expression for the perimeter of a rectangle with a length that is 5 inches longer than its width.

25. Write the expression for the perimeter of a rectangle with a length that is 4 centimeters longer than three times its width.

IV. Answer Key


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24. in

25. cm

©S R2a0I1s1c LKfuFtjaE CS2o7fBtRwpa5rpeR GL7LSCl.z R jAMlpli 3rai5g7hKtfsy ar3eVspegrpvmebdq.L 7 eMKavdWeg Qwmi7tBh9 OI3n9fKitnIiVtYes XA1ltgbefb6rUau V1T.0 Worksheet by Kuta Software LLC

MBF3C U2L1ws

©M b280f1W19 9KBuptkas DSWoLfQtnwnaJrFeM rLWLGCd.V O WAilUln Wr2iSguhltVsE drieasKeUrxvFehdq.9Combining Like Terms and the Distributive LawSimplify each expression.

1)

5

n

− 9

n

2)

−3

x

+ 9

x

3)

−2

m

+ 2

m

4)

1 + 10

r

+ 9

5)

10

b

+ 9

b

6)

8

n

− 8 −

n

7)

7

v

− 3 + 1 − 10

v

8)

−10 + 9

x

+ 7

x

+ 9

9)

2

n

+ 6

n

10)

−9 +

a

− 10

11)

−3(

−3

k

− 8) + 3

k

12)

7(

3

x

+ 8) + 3

13)

−

n

+

4(

4

n

− 1) 14)

−

x

−

7(

2 + 8

x

)

15)

p

−

p

(

−2

p

− 6) 16)

−6

x

(

−4 − 2

x

) + 3

x

2

17)

−8

m

(

1 + 5

m

) +

m

18)

−

n

+

8(

−

n

− 2)

19)

4(

8 − 2

b

) + 4 20)

−7 +

8(

1 + 8

r

)

21)

−

n

+

4

n

(

2 + 5

n

) 22)

−8(

x

− 6) + 7

23)

−8

a

2 +

4

a

(

2

a

+ 4) 24)

−3(

6 + 4

v

) − 2

25)

−(

x

− 3) + 8

x

26)

4

x

−

6(

x

+ 4)

27)

4

a

+

4(

7 + 3

a

) 28)

k

(

7

k

+ 3) − 3

k

2

29)

5

p

−

2

p

(

8 + 3

p

) 30)

−6

x

(

−5

x

− 6) − 8

x

2

31)

−

m

(

1 + 3

m

) − 3

m

2 32)

8

n

2 −

2

n

(

1 + 8

n

)

33)

5

r

−

6(

7

r

− 7) 34)

5

n

2 +

4

n

(

4

n

− 8)

35)

7

x

2 +

7

x

(

4 − 7

x

) 36)

−5

b

(

b

− 3) +

5(

b

− 6)

37)

−3

x

(

2 − 4

x

) −

6(

−7

x

+ 8) 38)

3

v

(

4

v

+ 8) −

2(

6

v

− 8)

39)

−5(

2 + 8

n

) +

8(

5 −

n

) 40)

−6(

4 − 6

k

) +

6(

5

k

− 1)

41)

6(

a

+ 5) +

5(

−8

a

− 1) 42)

−4(

−2 +

x

) +

6(

x

+ 4)

43)

8

x

(

1 + 8

x

) −

2

x

(

x

− 2) 44)

2

m

(

6

m

+ 2) −

3(

4

m

− 2)

45)

−4

n

(

1 + 3

n

) +

4(

3 − 2

n

) 46)

−

x

(

2 + 8

x

) −

5

x

(

1 − 6

x

)

47)

−6(

1 − 2

p

) −

6(

−2 − 3

p

) 48)

−4(

n

+ 1) +

4

n

(

7

n

+ 5)

49)

3(

−6

b

− 6) +

b

(

8

b

+ 4) 50)

5(

−3

r

− 7) −

7(

2

r

− 2)

©L D2a0w1Y1u 3KzuStvaP 3S4oLf9tqw1a5rweC mLILBCl.J t eAHlflV irsiSgYh2tGsE rrpeWsQePrWvAe5dd.l 3 QMMaMdyeH ewCi9tThs LIjnvfzinnPi6tWej gA4lGgpegbnrJak y1h.v Worksheet by Kuta Software LLC

Answers to Combining Like Terms and the Distributive Law1) −4

n

2) 6

x

3) 04)

10 + 10

r

5) 19

b

6)

7

n

− 87)

−3

v

− 2 8)

−1 + 16

x

9) 8

n

10)

−19 +

a

11)

12

k

+ 24 12)

21

x

+ 5913)

15

n

− 4 14)

−57

x

− 14 15)

7

p

+ 2

p

2

16)

24

x

+ 15

x

2 17)

−7

m

− 40

m

2 18)

−9

n

− 1619)

36 − 8

b

20)

1 + 64

r

21)

7

n

+ 20

n

2

22)

−8

x

+ 55 23) 16

a

24)

−20 − 12

v

25)

7

x

+ 3 26)

−2

x

− 24 27)

16

a

+ 2828)

4

k

2 + 3

k

29)

−11

p

− 6

p

2 30)

22

x

2 + 36

x

31)

−

m

− 6

m

2 32)

−8

n

2 − 2

n

33)

−37

r

+ 4234)

21

n

2 − 32

n

35)

−42

x

2 + 28

x

36)

−5

b

2 + 20

b

− 3037)

36

x

+ 12

x

2 − 48 38)

12

v

2 + 12

v

+ 16 39)

30 − 48

n

40)

−30 + 66

k

41)

−34

a

+ 25 42)

32 + 2

x

43)

12

x

+ 62

x

2 44)

12

m

2 − 8

m

+ 6 45)

−12

n

− 12

n

2 + 1246)

−7

x

+ 22

x

2 47)

6 + 30

p

48)

16

n

− 4 + 28

n

2

49)

−14

b

− 18 + 8

b

2 50)

−29

r

− 21

MBF3C U2L2 Mulitiplying Binomials

1

Topic : Expanding binomials

Goal : I can multiply binomials using the "double" distributive law, and simplify more complicated polynomial expressions.

Multiplying Binomials (aka double distributive law, aka FOIL)

When we have two sets of brackets being multiplied together, we must use the distributive law twice in order to expand.

Sometimes this is referred to as FOIL...

F O I L

FOIL isn't helpful when we have to expand something larger like the following where we will use double distributive to expand it out.

MBF3C U2L2 Mulitiplying Binomials

2

We will go through a couple more examples that you might run across...

Example 1 : Expand each of the following. Simplify when necessary.

a)

b)

c)

Practice Questions - Handout Page

MBF3C U2L2ws Multiplying Binomials Practice 1

Multiplying Binomials FOIL (Double Distributive) Practice Worksheet Find Each Product. 1. (x + 1)(x + 1) 2. (x + 1)(x + 2) 3. (x + 2)(x + 3) 4. (x + 3)(x + 2) 5. (x + 4)(x + 3) 6. (x - 6)(x + 2) 7. (x - 5)(x - 4) 8. (y + 6)(y + 5) 9. (2x + 1)(x + 2) 10. (y + 6)(3y + 2) 11. (2x + 1)(2x + 1) 12. (x + 5)(3x - 1) 13. (2x - 1)(x - 3) 14. (x + y)(x + y) 15. (3x + y)(x + y) 16. (2x + y)(2x – y) 17. (3x - y)(x + 2y) Find the area of each shape. 18. 19. x + 3 x + 6

x + 3 x + 3 20. 21. 2x + 7 x - 2 x + 2 4x – 2

MBF3C U2L2ws Multiplying Binomials Practice 2

22. Expand and simplify. 23. Expand and simplify.

24. Expand and simplify.

€

a)(x − 8)2

b)(5x − 3)2

c)(9x + 2)2

d)6x +12)2

e)(x + 7)2

€

a)(x +1)(x + 2) + 3(x − 2)

b)(x − 6)2 + (x + 2)2

c)(x − 7)(x + 7) − (x − 4)(x + 4)

d)(x + 5)(x + 3) + 2(x −1)2

e)3(x − 9)2 − 2(x − 3)(x + 6)

€

a)4(x − 5)(x + 5)b)7(x +1)(x − 6)c)3(2x −1)(3x + 4)d)2(x + 4)(x + 2)e)5(x − 9)(x − 5)

MBF3C U2L3 Quadratic Relationships

Topic : Quadratic Relationships

Goal : I understand the properties of a quadratic relationship and can recognize them in "real life" situations.

Quadratic RelationsThis is what a quadratic relationship looks like. One of the easiest to visualize is the height of a projectile

time height123456789

10

General Properties of a Quadratic Relations

• They have a high point or low point called a vertex.

• They are symmetric. A line called the axis of symmetry runs through the vertex. If you folded along that line the two sides would match up.

• The y-value of the vertex is considered the MAXIMUM or MINIMUM value of the relationship.

• The graph is always the same type of curve called a .• In a table of values, the second differences (see above) are always

constant.


Example The promotions manager of a new band is deciding how much to charge for concert tickets. She has calculated that if the tickets are $30 each, then 200 people will come to the concert. For every $1 increase in the price, 10 less people will come. Create a table to calculate how much should be charged to MAXMIZE the revenue from the ticket sales.


Total Money Vs. Ticket Price

Practice Questions - Page 102 #1-8

MBF3C U2L4 Transformations of the Parabola

Transformations of ParabolasWe are going to look at the equation

y = x2

This is basically a way of finding points on a graph. If we know the

x-coordinate, we can find the y-coordinate.

x y

By adding things to the equation, our basic function, we can move it around the grid.

The shape that is

graphed is called a parabola.

Properties of a parabola...

They have a lowest point (or highest if they are upside-down)This is referred to as the maximum or minimum value.

The turning point is called the vertex of the parabola.

A parabola is symmetric. The line of symmetry is a vertical linerunning right through the vertex - called the axis of symmetry.

Topic : Tranformations of Parabolas

Goal : I know how to graph the parabola y=x2 and how the graph is transformed when I change the equation.


Using graphing software, test out the following equations and see what transformation is involved...

y = x2

y = x2 + 2

y = x2 + 4

y = x2 - 2

y = x2 - 4

y = x2

y = (x+2)2

y = (x+4)2

y = (x-2)2

y = (x-4)2


y = (x-h)2+k

The general equation for a transformed parabola is

The value in the brackets moves it left or right.

The value on the end moves the parabola up or down.

The vertex of the parabola is (h, k) (notice that the h has the opposite sign as in the equation)

Now we'll try y = (x+3)2 - 4

The parabola has moved....

It's vertex is.....

Example 1. Describe the transformation of the parabola and state the vertex.

a) y = (x+2)2-7

b) y = (x-6)2

c) y = x2 + 5


Example 2. A parabola has been moved left 20 spaces and up two spaces. Write its equation.

Practice Questions - Page 114 #1-5, 8, 9

Example 3. What is the equation of the given parabola?

MBF3C U2L5 Graphing Tranformations of the Parabola

1

Transformations of Parabolas

x yy = x2

-3-2-10123

9410149


Goal : I can locate the vertex of a parabola using its equation and I can graph all other points on the parabola by using the vertex as my starting point.

Notice that for each HORIZONTAL

DISTANCE you move away from the

VERTEX, you move up that same distance SQUARED. This

happens whether you move left or right.

left/right up1 unit2 units3 units4 units5 units

.

.

.n units


2

Example 1. State the vertex of each parabola and then graph it using the pattern we discovered. Mark on the axis of symmetry for each.

a) y = (x+2)2-7 b) y = (x-6)2

c) y = x2 + 5 d) y = (x+4)2 + 1

d) y = (x-3)2 - 6 e) y = (x+1)2 + 3

Practice Questions - Worksheet

MBF3C&U2L5ws&Graphing&Transformations&of&the&Parabola&

&

1.&State&the&vertex&of&each&parabola&and&graph&on&the&grid&provided.&&

&

a)&&

€

y = x2& & & vertex&:&(&&&&&,&&&&&)&

&

b)&&

€

y = x2 −10& & vertex&:&(&&&&&,&&&&&)&

&

c)&&

€

y = x2 + 2& & vertex&:&(&&&&&,&&&&&)&

&

d)&&

€

y = (x − 5)2 − 8 & & vertex&:&(&&&&&,&&&&&)&

&

e)&&

€

y = (x +1)2 & & vertex&:&(&&&&&,&&&&&)&

&

f)&&

€

y = (x + 3)2 + 3& & vertex&:&(&&&&&,&&&&&)&

&

g)&&

€

y = (x − 4)2 − 5& & vertex&:&(&&&&&,&&&&&)&

&

&


&

a)&&

€

y = (x + 5)2 & & vertex&:&(&&&&&,&&&&&)&

&

b)&&

€

y = (x +1)2 −10 & & vertex&:&(&&&&&,&&&&&)&

&

c)&&

€

y = (x − 6)2 & & vertex&:&(&&&&&,&&&&&)&

&

d)&&

€

y = (x + 4)2 − 5& & vertex&:&(&&&&&,&&&&&)&

&

e)&&

€

y = (x −1)2 − 9 & & vertex&:&(&&&&&,&&&&&)&

&

f)&&

€

y = (x − 4)2 & & vertex&:&(&&&&&,&&&&&)&

&

g)&&

€

y = x2 + 2& & vertex&:&(&&&&&,&&&&&)&

&

&

3.&&State&the&equations&of&the&given&parabolas.&

a)&_________________________________&

&

b)&_________________________________&

&

c)&_________________________________&

&

d)&_________________________________&

&

e)&_________________________________&

&

f)&_________________________________&

&

g)&_________________________________&

&

&

&


1


x yy = x2

-3-2-10123

9410149


Goal : I can locate the vertex of a parabola using its equation and I can graph all other points on the parabola by using the vertex as my starting point.



VERTEX, you move up that same distance SQUARED. This



.

.

.n units


2


a) y = (x+2)2-7 b) y = (x-6)2

c) y = x2 + 5 d) y = (x+4)2 + 1

d) y = (x-3)2 - 6 e) y = (x+1)2 + 3


MBF3C&U2L5ws&Graphing&Transformations&of&the&Parabola&

&


&

a)&&

€

y = x2& & & vertex&:&(&&&&&,&&&&&)&

&

b)&&

€

y = x2 −10& & vertex&:&(&&&&&,&&&&&)&

&

c)&&

€

y = x2 + 2& & vertex&:&(&&&&&,&&&&&)&

&

d)&&

€

y = (x − 5)2 − 8 & & vertex&:&(&&&&&,&&&&&)&

&

e)&&

€

y = (x +1)2 & & vertex&:&(&&&&&,&&&&&)&

&

f)&&

€

y = (x + 3)2 + 3& & vertex&:&(&&&&&,&&&&&)&

&

g)&&

€

y = (x − 4)2 − 5& & vertex&:&(&&&&&,&&&&&)&

&

&


&

a)&&

€

y = (x + 5)2 & & vertex&:&(&&&&&,&&&&&)&

&

b)&&

€

y = (x +1)2 −10 & & vertex&:&(&&&&&,&&&&&)&

&

c)&&

€

y = (x − 6)2 & & vertex&:&(&&&&&,&&&&&)&

&

d)&&

€

y = (x + 4)2 − 5& & vertex&:&(&&&&&,&&&&&)&

&

e)&&

€

y = (x −1)2 − 9 & & vertex&:&(&&&&&,&&&&&)&

&

f)&&

€

y = (x − 4)2 & & vertex&:&(&&&&&,&&&&&)&

&

g)&&

€

y = x2 + 2& & vertex&:&(&&&&&,&&&&&)&

&

&

3.&&State&the&equations&of&the&given&parabolas.&

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b)&_________________________________&

&

c)&_________________________________&

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d)&_________________________________&

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e)&_________________________________&

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MBF3C U2L6 Graphing Stretches and Squishes

1


x yy = x2

-3-2-10123

9410149

Topic : Graphing Stretches and Squishes

Goal : I know how to change the equation of a parabola so that the shape of the parabola is transformed and I can graph this new parabola



VERTEX, you move up that same distance

SQUARED then multiplied by the 'a' - value. This



.

.

.n units


2


a) y = 3(x+1)2-10 b) y = (x-2)2

c) y = x2 + 5 d) y = 2(x+6)2 - 4

d) y = 4(x-3)2 - 10 e) y = 6(x-4)2

12

13


3


12

What happens if the "a-value" is negative?

Example 2. Graph the following parabolas. Pay attention to the direction of opening.

a) y = -3(x-2)2+10 b) y = (x+1)2-4

c) y =- x2 + 4 d) y = -2x2 - 4

d) y = -4(x-3)2 +10 e) y = 6x2

13

©Z w2A091Y13 7Kgubt7aV 3SooCfRtxwcaFr4e5 TLuLkCh.B F XA4lSl6 Xroi2gxhVtSs5 0rreQs5eTrIvWeDdb.D q KMYagdpeZ CwIiMtEhB mIQnIfAienHistbe0 NA4llgpeibCrTaG j2h.N Worksheet by Kuta Software LLC

MBF3C U2L6ws

©m o2g0D1o15 RKxuqtiaN uSvo1f0tEw2aqrdeN 1LDLvCz.L 8 YA6lql5 frUiJgrhUt0sw mrleksKevrAvDeGdn.8Graphing Parabolas in Vertex FormIdentify the vertex and direction of opening of each. Then sketch the graph.

1)

y

=

−

(

x

+ 1)2

2)

y

=

1

4

(

x

+ 6)2

3)

y

=

1

3

(

x

+ 3)2 4)

y

=

−

(

x

− 2)2 + 1

5)

y

=

x

2 − 16)

y

=

1

4

x

2 − 6

7)

y

=

−

(

x

+ 6)2 − 1 8)

y

=

2

(

x

+ 6)2 − 3

9)

y

=

2

(

x

− 3)2 10)

y

= −

x

2

11)

y

=

2

(

x

+ 5)2 − 4 12)

y

=

−1

4

(

x

+ 1)2 + 2

13)

y

=

−

(

x

− 6)2 + 4 14)

y

=

−2

(

x

+ 5)2 − 2

15)

y

=

(

x

+ 1)2 − 3 16)

y

=

1

3

(

x

− 4)2 + 4

17)

y

=

1

3

(

x

− 6)2 + 4 18)

y

=

−2

(

x

+ 4)2 − 4

19)

y

=

−

x

2 + 120)

y

=

−1

4

(

x

+ 5)2 + 2

21)

y

=

−1

4

(

x

+ 1)2 + 6 22)

y

=

−2

(

x

− 4)2

23)

y

=

−2

(

x

+ 1)2 − 6 24)

y

=

−1

4

(

x

− 5)2 + 3

25)

y

=

−

(

x

− 2)2 − 5 26)

y

=

1

2

(

x

− 3)2 + 2

27)

y

=

−1

4

(

x

− 1)2 + 4 28)

y

=

−3

(

x

− 1)2 − 6

29)

y

=

−1

3

(

x

− 2)2 + 3 30)

y

=

−

(

x

+ 3)2 − 6

31)

y

=

−

(

x

+ 2)2

32)

y

=

−1

3

(

x

+ 5)2 − 1

33)

y

=

−2

(

x

− 3)2 − 3 34)

y

=

−

(

x

− 1)2 − 1

35)

y

=

−2

(

x

− 6)2 − 1 36)

y

=

−1

4

(

x

− 5)2 + 1

37)

y

=

−

(

x

+ 6)2 + 4

collecting/combining like terms - pbworks

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