Unit 7 Algebra - mrvanscoyk.weebly.com
Transcript of Unit 7 Algebra - mrvanscoyk.weebly.com
Unit 7Quadratics
Tuesday May 11, 2021 Activator
What is the “Box Method?” It is a way to multiply with variables, an easier way to distribute.
Multiply (74)(392) = 29008
Multiply (70+4)(300+90+2) = 29008
Can you expand integers (normal numbers)? Yes
Why is the method used? to double distribute
Page #3Lesson 7.1
Today’s Objective
Students will be able to use the box method and combine like terms.
Unit 7Lesson 1
Today’s New Vocab (1 of 4)
+ 21,000 + 6,300 + 140
+ 1,200 + 360 +8
70+4
300 +90 +2Multiply using the “Box Method”
There must be a
sign in every box.Write down all of the boxes.
21,000 + 6,300 + 140 + 1,200 + 360 + 8 is 29,008
Combine like terms
Page #3Lesson 7.1
(2 of 4)
Combine Like Terms (CLT)
Definition
Example(s) Non-Example(s)
FactsTo write the answer
in an easier way.
• Used only with Add and Subtract
• The exponent does NOT change.
4𝑥2 + 3𝑥2 = 7𝑥2
4𝑥2 − 3𝑥2 = 𝑥24𝑥2 + 3𝑥3 ≠ 7𝑥5
Not the same exponent
Today’s New Vocab (3 of 4)Combine Like Terms: Does the exponent change?
Addition vs. No why? The exponent is multiplication.
(𝑥3) + (𝑥3) = 2𝑥3Subtraction
Exponent change? No
(5𝑥3) − (𝑥3) = 4𝑥3
Exponent change? No
𝑥3 + 𝑥3 = 2𝑥3 5𝑥3 − 𝑥3 = 4𝑥3
Why? We did not multiply a variable by a variable.
Today’s New Vocab (4 of 4)Combine Like Terms Will the answer have an equal sign? No Why? No equal sign in the question.
Line 1: (7𝑥3+ 3𝑥2) - (9x - 5𝑥2)
Line 2: 7𝑥3+ 3𝑥2 - 9x + 5𝑥2
Line 4: 7𝑥3 + 8𝑥2 - 9x
Distribute
Combine Like Terms
Line 3: 7𝑥3+ 3𝑥2 + 5𝑥2 - 9x Commute
Page #4Lesson 7.1
Group Work Questions
#1#2#3#4#5
#6#7#8#9#10
7, 2, 4
8, 1, 69, 3, 510, 2, 411, 1, 6
7, 3, 5
8, 2, 49, 1, 610, 3, 511, 2, 4
Page #1-2Lesson 7.1
Tuesday May 11, 2021 Work Period
(2𝑥2 + 6x + 5 ) - (6𝑥2 +3x + 5)
If A=(2𝑥2 + 6x + 5 ) and B=(6𝑥2 +3x + 5), what is A-B?
2𝑥2 + 6x + 5 - 6𝑥2 - 3x - 52𝑥2- 6𝑥2 + 6x - 3x + 5 - 5
-4𝑥2 + 3x + 0
-4𝑥2 + 3x
Distribute
Commutative
Combine like Terms
Page #4Lesson 7.1
.
Tuesday May 11, 2021 Exit TicketWhat is the sum of 8𝑥2 - x + 4 and x - 5 ?
(8𝑥2 - x + 4) + (x – 5) 8𝑥2 - x + 4 + x - 5
8𝑥2 + 0x - 1 8𝑥2 - 1
8𝑥2 - x + x + 4 - 5
Distribute
Commutative
Combine like Terms
Page #4 Lesson 7.1
Friday May 14, 2021 Activator
Monomial
2𝑥3(5𝑥3) 2𝑥3 + 5𝑥3
Simplify the expressionsMonomial Monomial Monomial
10𝑥6
7𝑥32∙ 𝑥3(5)(𝑥3)2∙ 5 𝑥3(𝑥3) Does the exponent change?
No Why? Addition and subtraction does not change exponents.
Page #7Lesson 7.2
Today’s Objective
Students will be able to multiply polynomials.
Unit 7Lesson 2
(1 of 4)
Monomial
Definition
Example(s) Non-Example(s)
Facts
• has one sign and/or number
and/or variable
-1 -1𝑥2
3x 5 8𝑥4x - 1 2x+4
−2𝑥2 + 3𝑥
Is one part of an expression
Page #7Lesson 7.2
Today’s New Vocab (2 of 4)Simplify the expression
x (x – 4)
x
x – 4
Can this be graphed? NoWhy? The variable needs distributed.
𝑥2 - 4x
A sign (±) 𝑚𝑢𝑠𝑡 𝑔𝑜 𝑖𝑛 𝑒𝑎𝑐ℎ 𝑏𝑜𝑥.
Are these like terms? ______Same variable & same exponent
No+ 𝒙𝟐
- 4x
Page #7Lesson 7.2
Today’s New Vocab (3 of 4)Simplify the expression
x (𝑥2 + x - 4) Write all boxes down
A sign (±) 𝑚𝑢𝑠𝑡 𝑔𝑜 𝑖𝑛 𝑒𝑎𝑐ℎ 𝑏𝑜𝑥.
Are these like terms? ______Same variable & same exponent
+𝒙𝟑 +𝒙𝟐 -4xx
𝑥2 + x - 4
𝑥3 + 𝑥2 - 4x
No Can this be graphed? NoWhy? The variable needs distributed.
Today’s New Vocab (4 of 4)Simplify the expression
(x + 3)(x - 4) Write all boxes down
A sign (±) 𝑚𝑢𝑠𝑡 𝑔𝑜 𝑖𝑛 𝑒𝑎𝑐ℎ 𝑏𝑜𝑥.
+𝒙𝟐 -4x
+3x -12
x + 3
x -4 𝑥2 − 4𝑥 + 3𝑥 − 12
𝑥2 − 1𝑥 − 12Can this be graphed? YesWhy? The variables are in ( ).
Page #8Lesson 7.2
Group Work Questions
#1#2#3#4#5
#6#7#8#9#10
1, 3, 9, 17
2, 4, 10, 18 5, 3, 11, 191, 4, 12, 202, 3, 9, 17
5, 4, 10, 181, 3, 11, 19 2, 4, 12, 20
5, 3, 9, 171, 4, 10, 18
Page #5-6Lesson 7.2
Friday May 14, 2021 Work Period
.
The expression (𝑥 − 6)2 is equivalent to
+ 𝒙𝟐 - 6x
- 6x + 36
𝑥 − 6𝑥
−6
Write all boxes down
𝑥2 − 6𝑥 − 6𝑥 + 36
𝑥2 − 12𝑥 + 36
LikeTerms
Page #8Lesson 7.2
Friday May 14, 2021 Exit Ticket F(x) = (𝑥 − 6)2
x G(x)
4 4
5 1
6 0
7 1
G(x) = 𝑥2 − 12𝑥 + 36 𝑎𝑛𝑑
Does G(x) and F(x) have an infinite (ALL) number of solutions?
Yes, because it is the same line.
Tuesday May 18, 2021 Activator Simplify the following expressions. Are they equivalent? YES
x(x) -2𝑥2 + 3𝑥2
𝑥2 𝑥2Does the exponent change? Does the exponent change?
Yes. Why? Multiplying variables
No Why? Combining variables
Page #11Lesson 7.3
Today’s Objective
Students will be able to multiply binomials.
Unit 7Lesson 3
Today’s New Vocab (1 of 4)Axis of Symmetry –
x = lines that splits the graph in half.
Root, Zero, or Solutions –
Where the graph crosses the x-axis.
Vertex – The maximum or Minimum point on the graph.
Page #11Lesson 7.3
.
When x = -1 and x = 3, write the factors
(x + 1) ( x - 3) = 0 (x + 1) = 0 ( x - 3) = 0
x = -1 x = 3 +1 +1 -3 -3
Solution
Factor x - 3 = 0 x + 1 = 0
Factor
Solution
Today’s New Vocab (2 of 4)
To write the factors, you need to sign switch from the solutions.
Page #11Lesson 7.3
Today’s New Vocab (3 of 4)
+ 𝒙𝟐 - 3x
+ 1x -3
Determine the product of the following expression.
𝑥 − 3𝑥
+1
Write all boxes down
𝑥2 + 1𝑥 − 3𝑥 − 3
𝑥2 − 2𝑥 − 3
LikeTerms
(x + 1)(x - 3)
Page #11Lesson 7.3
Today’s New Vocab (4 of 4)Graph the polynomial f(x) = (x + 1)(x − 3)
x f(x)
-1 0
0 -3
1 -4
2 -3
3 0
BOX the Roots
X = _____X = _____
-13
(-1,0) (3,0)Page #12
Lesson 7.3
Group Work Questions
#1#2#3#4#5
#6#7#8#9#10
3, 4, 1
3, 4, 2 3, 4, 1
3, 4, 1
3, 4, 1
3, 4, 13, 4, 2
3, 4, 2
3, 4, 2
3, 4, 2
Page #9-10Lesson 7.3
+ 2𝒙𝟐 - 4x
+ 4x - 8
Determine the product of the following expression.
2𝑥 − 4𝑥
+2
Write all boxes down
2𝑥2 − 4𝑥 + 4𝑥 − 8
2𝑥2 − 8
LikeTerms
(x + 2)(2x - 4)
Tuesday May 18, 2021 Work Period
Page #12Lesson 7.3
Tuesday May 18, 2021 Exit Ticket F(x) = (x + 2)(2x - 4) G(x) = 2𝑥2− 8 𝑎𝑛𝑑
Is (-2,0) a solution to the system?
Yes, because it is on both lines and both tables.
x F(x)
-2 0
-1 -6
0 -8
1 -6
2 0
Graph the polynomial f(x) = (x + 1)(x +4)
Write the Vertex (1, -4)
Is the Vertex a Minimum or Maximum?
Why? Vertex is at the bottom of the graph.
Minimum
Monday May 24, 2021 Activator
Page #15Lesson 7.4
Today’s Objective
Students will be able to graph quadratics.
Unit 7Lesson 4
Today’s New Vocab (1 of 4)Factor the polynomial f(x) = 𝑥2+ 5x + 4
x f(x)
-4 0
-3 -2
-2 -2
-1 0
BOX the Solutions
Can you get the factors from the graph? Yes How?Change the signs on the Solutions.
f(x) = (x + 1) ( x + 4) Page #15
Lesson 7.4
.
Solve for x when 𝑥2 + 5x + 4 = 0 ?
(x + 1) ( x + 4) = 0
(x + 1) = 0 ( x + 4) = 0
x = -1 x = -4 -1 -1 -4 -4
This graph will cross the x-axis at (-4,0) and (-1,0).
Set both parentheses equal to zero.
Solution
Factor x + 4 = 0 x + 1 = 0
Factor
Solution
Today’s New Vocab (2 of 4)
Page #15Lesson 7.4
Today’s New Vocab (3 of 4)Graph the polynomial f(x) = −2(𝑥 − 1)2
x f(x)
-1 -8
0 -2
1 0
2 -2
3 -8
BOX the Zero’s
X = ___1(1,0)
Can the vertex also be a zero?
YESPage #15
Lesson 7.4
Today’s New Vocab (4 of 4)Graph the polynomial f(x) = 2𝑥3 − 12𝑥2 + 10𝑥
x f(x)
0 0
1 0
5 0
BOX the Zero’s
X = ___
X = ___
Write the solutions.
X = ___0
15 Page #16
Lesson 7.4
Group Work Questions
#1#2#3#4#5
#6#7#8#9
#10
10, 16, 2, 5
11, 17, 3, 7 12, 18, 4, 610, 16, 2, 511, 17, 3, 7
12, 18, 4, 6
10, 16, 2, 5 11, 17, 3, 7 12, 18, 4, 610, 16, 2, 5
Page #13-14Lesson 7.4
Monday May 24, 2021 Work PeriodCompare the graph of f(x) = 𝑥2 to the graph
of g(x) = (𝑥 − 2)2 + 3. Which two directions did the g(x) shift(move)? 2 right and 3 up
x f(x)
-2 4
0 0
2 4
x g(x)
0 7
2 0
4 7
f(x) g(x)
Monday May 24, 2021 Exit TicketWhat is f(6) – g(6)? 36 – 19 = 17
f(x) = 𝑥2 g(x) = (𝑥 − 2)2 + 3
f(6) = 36
g(6) = (6 − 2)2 + 3f(6) = (6)2
g(6) = (4)2 + 3
g(6) = 16 + 3
g(6) = 19
Show your work. Page #16
Lesson 7.4
Thursday May 27, 2021 Activator
1
2
How do you graph radical equations?
Page #19Lesson 7.5
x
2 steps
Today’s Objective
Students will be able to graph quadratic (radical) equations.
Unit 7Lesson 5
(1 of 4)
Radical/Root
Definition
Example(s) Non-Example(s)
Facts
Page #19Lesson 7.5
• Opposite of an exponent• Fractional • exponent
𝑥
9 = 3 16 = 4
𝑥12 𝑥2
x𝑥3
A radical(root) is an operation to remove (undo) an exponent.
Today’s New Vocab (2 of 4)Perfect Squares
9 = 3
16 = 4
Calculate the (square) roots.
25 = 5
36 = 6
49 = 7
100 = 10
144 = 12
Today’s New Vocab (3 of 4)
x f(x)
0 0
1 1
4 2
9 3
Graph the function f(x) = 𝑥How is this calculated?
f(x) = 𝑥
f(9) = 9
f(9) = 3 Page #19Lesson 7.5
Today’s New Vocab (4 of 4)
x f(x)
-5 3
-4 4
-1 5
4 6
Graph the function f(x) = 𝑥 + 5 + 3
How did the graph shift?
5 units left3 units up Page #20
Lesson 7.5
Group Work Questions
#1#2#3#4#5
#6#7#8#9#10
1,5,7,11,13
1,6,9,12,13
Page #17-18Lesson 7.5
1,5,7,11,13
1,6,9,12,13 1,5,7,11,13
1,6,9,12,13 1,5,7,11,13
1,6,9,12,13 1,5,7,11,131,6,9,12,13
Thursday May 27, 2021 Work Period
.
Graph f(x) = 𝑥 + 2 over the domain −2 ≤ x ≤ 7.
x f(x)
-2 0
-1 1
2 2
7 3
Intervals have no ARROWS.
starts. ends.Is there shading on the graph?
No, it is an equation.
Page #20Lesson 7.5
Thursday May 27, 2021 Exit Ticket
.
Evaluate f(-2) and f(7) when f(x) = 𝑥 + 2.
f(x) = 𝑥 + 2
f(-2) = (−2) + 2
f(-2) = 0
f(-2) = 0(-2,0) is a point
on the line.
f(x) = 𝑥 + 2
f(7) = (7) + 2
f(7) = 9
f(7) = 3(7,3) is a point
on the line.
Thursday June 3, 2021 Activator
Page #23Lesson 7.6
Can you take the (square root) of a negative number? NoWhy? Two of the same numberscannot multiply to be negative
Calculate −9
Error: non-real (irrational)
(3) (3) = +9(-3)(-3)= +9
Today’s Objective
Students will be able to solving equations with
exponents using radicals.
Unit 7Lesson 6
Today’s New Vocab (1 of 4)
Page #23Lesson 7.6
How do you solve radical equations?
You can (square) root both sides.
Solve. 𝑥2 = 36
x = 6
Check your
Work.
𝑥2 = 36
(6)2= 36
36 = 36
Yes, x = 6 is a solution.
Today’s New Vocab (2 of 4)
Page #23Lesson 7.6
However, is x = 6 the only solution? No
𝑥2 = 36
(−6)2= 36
36 = 36
x = -6 is
also a solution.
Check on the calculator.
Make a table for f(x) = 𝑥2 - 36
x f(x)
-6 0
6 0
Today’s New Vocab (3 of 4)
Page #23Lesson 7.6
Solve the quadratic (number exp.) equation.
𝑥2 − 5 = 44
𝑥2 = 9
𝑥2 + 7 = 16
𝑥2 = 49
x = 3
x = −3
-7 -7
x = 7
x = −7
+5 +5x y
7 0
-7 0x y
3 0
-3 0
Today’s New Vocab (4 of 4)
Page #24Lesson 7.6
Solve the quadratic (number exp.) equation.𝑥2
3= 27
4𝑥2 − 3 = 97
x y
9 0
-9 0x y
5 0
-5 0
4𝑥2 = 100
X = ±5
𝑥2 = 25
+3 +3
÷ 4 ÷ 4 𝑥2 = 81
X = ±9
(3) (3)
Group Work Questions
#1#2#3#4#5
#6#7#8#9#10
1A,3A,6A,7A
Page #21-22Lesson 7.6
1A,4A,5A,8A
1A,3A,6A,7A
1A,4A,5A,8A
1A,3A,6A,7A
1A,4A,5A,8A
1A,3A,6A,7A
1A,4A,5A,8A
1A,3A,6A,7A
1A,4A,5A,8A
Thursday June 3, 2021 Work Period
.
A landscaper is creating a square flower bed such that the length is L, feet. The area of the flower bed is 81 square feet. Write and solve an equation to determine length.
Let L = Length 𝐿2 = 81
𝐿 = 9The length of the
flower bed is 9 feet.
How many 1-foot stones are needed for the perimeter? 9+9+9+9 = 36 stones Page #24
Lesson 7.6
Thursday June 3, 2021 Exit Ticket
.
Determine the cost of the garden with tax.
Page #24Lesson 7.6
Stones (36) at $1.95 each
Flowers (25) at $3.95 eachMulch (6) at $3.33 each
Weed Barrier (2) rolls at $15.99 each
$220.91 (1.08) = $238.58Tax Final Cost
Monday June 7, 2021 Activator
Page #27Lesson 7.7
Solve for x. How do you remove the root? ( )2
4 = X
( )2 ( )23 = 𝑥 + 5
9 = x + 5-5 -5
Check your work.
3 = (4) + 5
3 = 93 = 3 Yes
Today’s Objective
Students will be able to graph complex radical equations.
Unit 7Lesson 7
Today’s New Vocab (1 of 4)
Page #27Lesson 7.7
Graph g(x) = 3 − 𝑥 + 5. x g(x)
-5 3
-4 2
-1 1
4 0
Is (4,0) a root?
Yes, it is on the table next
to a zero.
Today’s New Vocab (2 of 4)
Page #27Lesson 7.7
Evaluate g(4) when g(x) = 3 − 𝑥 + 5.
g(4) = 3 − (4) + 5
g(4) = 3 − 9
g(4) = 3 − 3
g(4) = 0
x g(x)
-2 1.26
4 0
Is g(4) rational or irrational? Rational b/c
9 𝑖𝑠 𝑝𝑒𝑟𝑓𝑒𝑐𝑡.
Is g(-2) rational? NoIt has a decimals on the table.
Today’s New Vocab (3 of 4)
Page #27Lesson 7.7
Graph f(x) = 1
3𝑥 + 4.
x f(x)
-4 0
5 1
Is (-4,0) a root?
Yes, it is on the table next
to a zero.
Today’s New Vocab (4 of 4)
Page #28Lesson 7.7
Evaluate f(-4) when f(x) = 1
3𝑥 + 4.
f(-4) = 1
30
f(-4) = 0
x g(x)
-4 0
-2 0.47
Is f(-4) rational or irrational? Rational b/c
0 𝑖𝑠 𝑝𝑒𝑟𝑓𝑒𝑐𝑡.
Is f(-2) rational? No,It has a decimals on the table.
f(-4) = 1
3(−4) + 4
Group Work Questions
#1#2#3#4#5
#6#7#8#9#10
Page #25-26Lesson 7.7
Pick any 3 graphs.Pick any 3 graphs.Pick any 3 graphs.
Pick any 3 graphs.Pick any 3 graphs.
Monday June 7, 2021 Work Period
.
The number of people, p involved in recycling in a community is
modeled by the function p = 90 3𝑥 + 400, where x is the number
of months the recycling plant has been open. How people wereinvolved starting out? After 3 months? After 12 months?
x P(x)
0 400
3 670
12 940
Does this graph indicate growth or decay of this
recycling program? GrowthWhy? The people helping
are increasing.
Monday June 7, 2021 Exit Ticket
.
Page #28Lesson 7.7
The number of people, p involved in recycling in a
community is modeled by the function p = 90 3𝑥 + 400.
How many people will be helping after 4 years(48 months)?
p(x) = 90 3𝑥 + 400
p(48) = 90 3(48) + 400
p(48) = 90 144 + 400
p(48) = 90(12) + 400
p(48) = 880
x p(x)
48 880
The more helping hands the better.
Thursday June 10, 2021 Activator
#5: Evaluate the function when x = 8.
Page #29Unit 7 Review
Today’s Objective
Students will be able to review objectives from Unit 7.
Unit 7Review
Today’s New Vocab (1 of 4)#10: What is the product of (3x-3) and (x+1)?
Product means multiply
Page #31Unit 7 Review
Today’s (2 of 4)#11: Make a table, graph.
12. Write the Solutions
x = -1 and x = 1
Page #31Unit 7 Review
Page #32Unit 7 Review
Today’s New Vocab (3 of 4)#17. Solve the quadratic equation for x.(𝑥 + 3)2 = 49
(x + 3) = 7x + 3 = 7
x = 4-3 -3
(𝑥 + 3)2 = 49
(x + 3) = -7x + 3 = -7
x = -10-3 -3
Today’s New Vocab (4 of 4)#14: Simplify the expression. Page #31
Unit 7 Review
GWQ’s Skip #9
#1#2#3#4#5
#6#7#8#9#10
1,6,7,8
2,3,4,19 1,15,16,182,6,7,81,3,4,19
2,15,16,18
1,6,7,8 2,3,4,19 1,15,16,182,3,4,19
Page #29-32Unit 7 Review
#9: What is equivalent to 2𝑥2(4x + 5)?
𝐼𝑠 2𝑥2
𝑎 𝑓𝑎𝑐𝑡𝑜𝑟? YesPage #30
Unit 7 Review
Thursday June 10, 2021 Exit Ticket
Monday June 14, 2021 Activator • Please take out your Homework • Please take out your Classwork• Unit #7 Test: 10% Question #19• No work needed on #2• You must receive at least a 65%, or you
must stay after school to complete it.
• Tutoring: Mon 2:45-4:15, Wed 2:45-4:15