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 Terms
Simplify the following
What does simplify mean?
put together anything that is the same (aka collect like terms
What are like terms?
any terms that share the exact same variables - this means the
variables must have the same exponents.
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
3(5+2)
The Distributive Law
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)
c)
Practice Questions - Handout Page
MBF3C U2L1ws Combining Like Terms
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
www.mathworksheetsgo.com
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
www.mathworksheetsgo.com
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
www.mathworksheetsgo.com
MBF3C U2L1ws
Answers to Combining Like Terms and the Distributive Law
Combining Like Terms and the Distributive Law
1) −4n
4) 10 + 10r
7) −3v − 2
10) −19 + a
13) 15n − 4
16) 24 x + 15 x 2
19) 36 − 8b
22) −8 x + 55
25) 7 x + 3
28) 4k 2 + 3k
31) −m − 6m 2
34) 21n 2 − 32n
37) 36 x + 12 x 2 − 48
40) −30 + 66k
43) 12 x + 62 x 2
46) −7 x + 22 x 2
49) −14b − 18 + 8b 2
©M b280f1W19 9KBuptkas DSWoLfQtnwnaJrFeM rLWLGCd.V O WAilUln Wr2iSguhltVsE drieasKeUrxvFehdq.9
Simplify each expression.
1) 5n − 9n
2) −3 x + 9 x
3) −2m + 2m
4) 1 + 10r + 9
5) 10b + 9b
6) 8n − 8 − n
7) 7v − 3 + 1 − 10v
8) −10 + 9 x + 7 x + 9
9) 2n + 6n
10) −9 + a − 10
11) −3(−3k − 8) + 3k
12) 7(3 x + 8) + 3
13) −n + 4(4n − 1)
14) − x − 7(2 + 8 x)
15) p − p(−2 p − 6)
16) −6 x(−4 − 2 x) + 3 x 2
17) −8m(1 + 5m) + m
18) −n + 8(−n − 2)
19) 4(8 − 2b) + 4
20) −7 + 8(1 + 8r)
21) −n + 4n(2 + 5n)
22) −8( x − 6) + 7
23) −8a + 4a(2a + 4)
24) −3(6 + 4v) − 2
25) −( x − 3) + 8 x
26) 4 x − 6( x + 4)
27) 4a + 4(7 + 3a)
28) k(7k + 3) − 3k 2
29) 5 p − 2 p(8 + 3 p)
30) −6 x(−5 x − 6) − 8 x 2
31) −m(1 + 3m) − 3m 2
32) 8n 2 − 2n(1 + 8n)
33) 5r − 6(7r − 7)
34) 5n 2 + 4n(4n − 8)
35) 7 x 2 + 7 x(4 − 7 x)
36) −5b(b − 3) + 5(b − 6)
37) −3 x(2 − 4 x) − 6(−7 x + 8)
38) 3v(4v + 8) − 2(6v − 8)
39) −5(2 + 8n) + 8(5 − n)
40) −6(4 − 6k) + 6(5k − 1)
41) 6(a + 5) + 5(−8a − 1)
42) −4(−2 + x) + 6( x + 4)
43) 8 x(1 + 8 x) − 2 x( x − 2)
44) 2m(6m + 2) − 3(4m − 2)
45) −4n(1 + 3n) + 4(3 − 2n)
46) − x(2 + 8 x) − 5 x(1 − 6 x)
47) −6(1 − 2 p) − 6(−2 − 3 p)
48) −4(n + 1) + 4n(7n + 5)
49) 3(−6b − 6) + b(8b + 4)
50) 5(−3r − 7) − 7(2r − 2)
2
©S R2a0I1s1c LKfuFtjaE CS2o7fBtRwpa5rpeR GL7LSCl.z R jAMlpli 3rai5g7hKtfsy ar3eVspegrpvmebdq.L 7 eMKavdWeg Qwmi7tBh9 OI3n9fKitnIiVtYes XA1ltgbefb6rUau V1T.0
Worksheet by Kuta Software LLC
2) 6 x
5) 19b
8) −1 + 16 x
11) 12k + 24
14) −57 x − 14
17) −7m − 40m 2
20) 1 + 64r
23) 16a
26) −2 x − 24
29) −11 p − 6 p 2
32) −8n 2 − 2n
35) −42 x 2 + 28 x
38) 12v 2 + 12v + 16
41) −34a + 25
44) 12m 2 − 8m + 6
47) 6 + 30 p
50) −29r − 21
©L D2a0w1Y1u 3KzuStvaP 3S4oLf9tqw1a5rweC mLILBCl.J t eAHlflV irsiSgYh2tGsE rrpeWsQePrWvAe5dd.l 3 QMMaMdyeH ewCi9tThs LIjnvfzinnPi6tWej gA4lGgpegbnrJak y1h.v
3) 0
6) 7n − 8
9) 8n
12) 21 x + 59
15) 7 p + 2 p 2
18) −9n − 16
21) 7n + 20n 2
24) −20 − 12v
27) 16a + 28
30) 22 x 2 + 36 x
33) −37r + 42
36) −5b 2 + 20b − 30
39) 30 − 48n
42) 32 + 2 x
45) −12n − 12n 2 + 12
48) 16n − 4 + 28n 2
Worksheet by Kuta Software LLC
MBF3C U2L2 Mulitiplying Binomials
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.
1
MBF3C U2L2 Mulitiplying Binomials
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
2
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
22. Expand and simplify.
2
23. Expand and simplify.
a)(x − 8) 2
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)
b)(5x − 3) 2
c)(9x + 2) 2
d)6x + 12) 2
e)(x + 7) 2
€
24. Expand and simplify.
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)
€
€
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 Relations
This is what a quadratic relationship looks like. One of the easiest
to visualize is the height of a projectile
time
height
1
2
3
4
5
6
7
8
9
10
General Properties of a Quadratic Relations
• The graph is always the same type of curve called a
.
• In a table of values, the second differences (see above) are always
constant.
• 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.
MBF3C U2L3 Quadratic Relationships
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.
MBF3C U2L3 Quadratic Relationships
Total Money Vs. Ticket Price
Practice Questions - Page 102 #1-8
MBF3C U2L4 Transformations of the Parabola
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.
Transformations of Parabolas
We 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
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 line
running right through the vertex - called the axis of symmetry.
By adding things to the equation, our basic function, we can move it
around the grid.
MBF3C U2L4 Transformations of the Parabola
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
MBF3C U2L4 Transformations of the Parabola
Now we'll try y = (x+3)2 - 4
The parabola has moved....
It's vertex is.....
The general equation for a transformed parabola is
y = (x-h)2+k
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)
Example 1.
a) y = (x+2)2-7
b) y = (x-6)2
c) y = x2 + 5
Describe the transformation of the parabola and state
the vertex.
MBF3C U2L4 Transformations of the Parabola
Example 2.
A parabola has been moved left 20 spaces and up
two spaces. Write its equation.
Example 3.
What is the equation of the given parabola?
Practice Questions - Page 114 #1-5, 8, 9
MBF3C U2L4 Transformations of the Parabola
MBF3C U2L5 Graphing Tranformations of the Parabola
Topic :
Tranformations of Parabolas
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.
Transformations of Parabolas
y = x2
y
x
-3
-2
-1
0
1
2
3
left/right
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.
9
4
1
0
1
4
9
up
1 unit
2 units
3 units
4 units
5 units
.
.
.
n units
1
MBF3C U2L5 Graphing Tranformations of the Parabola
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
2
MBF3C&U2L5ws&Graphing&Transformations&of&the&Parabola&
&
1.&State&the&vertex&of&each&parabola&and&graph&on&the&grid&provided.&&
&
a)&& y = x 2 & &
&
vertex&:&(&&&&&,&&&&&)&
&
b)&& y = x 2 − 10 &
&
vertex&:&(&&&&&,&&&&&)&
€ &
c)&& y = x 2 + 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&:&(&&&&&,&&&&&)&
€ &
&
€2.&State&the&vertex&of&each&parabola&and&graph&on&the&grid&provided.&&
&
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 = x 2 + 2 &
&
vertex&:&(&&&&&,&&&&&)&
€&
&
€3.&&State&the&equations&of&the&given&parabolas.&
a)&_________________________________&
&
b)&_________________________________&
&
c)&_________________________________&
&
d)&_________________________________&
&
e)&_________________________________&
&
f)&_________________________________&
&
g)&_________________________________&
&
&
&
MBF3C U2L5 Graphing Tranformations of the Parabola
Topic :
Tranformations of Parabolas
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.
Transformations of Parabolas
y = x2
y
x
-3
-2
-1
0
1
2
3
left/right
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.
9
4
1
0
1
4
9
up
1 unit
2 units
3 units
4 units
5 units
.
.
.
n units
1
MBF3C U2L5 Graphing Tranformations of the Parabola
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
2
MBF3C&U2L5ws&Graphing&Transformations&of&the&Parabola&
&
1.&State&the&vertex&of&each&parabola&and&graph&on&the&grid&provided.&&
&
a)&& y = x 2 & &
&
vertex&:&(&&&&&,&&&&&)&
&
b)&& y = x 2 − 10 &
&
vertex&:&(&&&&&,&&&&&)&
€ &
c)&& y = x 2 + 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&:&(&&&&&,&&&&&)&
€ &
&
€2.&State&the&vertex&of&each&parabola&and&graph&on&the&grid&provided.&&
&
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 = x 2 + 2 &
&
vertex&:&(&&&&&,&&&&&)&
€&
&
€3.&&State&the&equations&of&the&given&parabolas.&
a)&_________________________________&
&
b)&_________________________________&
&
c)&_________________________________&
&
d)&_________________________________&
&
e)&_________________________________&
&
f)&_________________________________&
&
g)&_________________________________&
&
&
&
MBF3C U2L6 Graphing Stretches and Squishes
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
Transformations of Parabolas
y=
x2
y
x
-3
-2
-1
0
1
2
3
left/right
Notice that for each
HORIZONTAL
DISTANCE you move
away from the
VERTEX, you move up
that same distance
SQUARED then
multiplied by the
'a' - value. This
happens whether you
move left or right.
9
4
1
0
1
4
9
up
1 unit
2 units
3 units
4 units
5 units
.
.
.
n units
1
MBF3C U2L6 Graphing Stretches and Squishes
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 = 3(x+1)2-10
b) y = 1 (x-2)2
2
c) y = 1 x2 + 5
3
d) y = 2(x+6)2 - 4
d) y = 4(x-3)2 - 10
e) y = 6(x-4)2
2
MBF3C U2L6 Graphing Stretches and Squishes
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 = 1(x+1)2-4
2
c) y =- 1x2 + 4
3
d) y = -2x2 - 4
d) y = -4(x-3)2 +10
e) y = 6x2
Practice Questions - Worksheet
3
MBF3C U2L6ws
Graphing Parabolas in Vertex Form
©m o2g0D1o15 RKxuqtiaN uSvo1f0tEw2aqrdeN 1LDLvCz.L 8 YA6lql5 frUiJgrhUt0sw mrleksKevrAvDeGdn.8
Identify the vertex and direction of opening of each. Then sketch the graph.
1) y = −( x + 1) 2
3) y =
2) y =
1
( x + 6) 2
4
4) y = −( x − 2) + 1
2
1
( x + 3) 2
3
5) y = x 2 − 1
6) y =
1 2
x −6
4
7) y = −( x + 6) − 1
2
8) y = 2( x + 6) − 3
9) y = 2( x − 3)
2
10) y = − x 2
2
11) y = 2( x + 5) 2 − 4
12) y = −
13) y = −( x − 6) 2 + 4
14) y = −2( x + 5) 2 − 2
15) y = ( x + 1) 2 − 3
17) y =
1
( x − 6) 2 + 4
3
19) y = − x 2 + 1
21) y = −
16) y =
1
( x − 4) 2 + 4
3
18) y = −2( x + 4) 2 − 4
20) y = −
1
( x + 1) 2 + 6
4
1
( x + 1) 2 + 2
4
1
( x + 5) 2 + 2
4
22) y = −2( x − 4)
23) y = −2( x + 1) − 6
24) y = −
25) y = −( x − 2) − 5
26) y =
2
2
2
1
( x − 5) 2 + 3
4
1
( x − 3) 2 + 2
2
27) y = −
1
( x − 1) 2 + 4
4
28) y = −3( x − 1) − 6
29) y = −
1
( x − 2) 2 + 3
3
30) y = −( x + 3) − 6
2
2
31) y = −( x + 2) 2
32) y = −
33) y = −2( x − 3) 2 − 3
34) y = −( x − 1) 2 − 1
35) y = −2( x − 6) 2 − 1
36) y = −
1
( x + 5) 2 − 1
3
1
( x − 5) 2 + 1
4
37) y = −( x + 6) 2 + 4
©Z w2A091Y13 7Kgubt7aV 3SooCfRtxwcaFr4e5 TLuLkCh.B F XA4lSl6 Xroi2gxhVtSs5 0rreQs5eTrIvWeDdb.D q KMYagdpeZ CwIiMtEhB mIQnIfAienHistbe0 NA4llgpeibCrTaG j2h.N
Worksheet by Kuta Software LLC