Download Algebra I - Windsor C

Algebra I
Chapter 8/9 Notes
Part II
8-5, 8-6, 8-7, 9-2, 9-3
Section 8-5: Greatest Common Factor,
Day 1
Factors –
Factoring –
Standard Form
3x - 9x
2
Factored Form
3(x - 3)
Section 8-5: Greatest Common Factor,
Day 1
Factors – the numbers, variables, or expressions
that when multiplied together produce the
original polynomial
Factoring – The process of finding the factors of
a polynomial
Standard Form
Factored Form
3x - 9x
2
3(x - 3)
Section 8-5: GCF, Day 1
Greatest Common Factor (GCF): The largest
factor in a polynomial. Factor this out FIRST in
every situation
Ex ) Factor out the GCF
1) 27y2 +18y
2) -4a2b -8ab2 + 2ab
3) 7u2t 2 + 21ut 2 - ut
4) 15w – 3v
Section 8-5: Grouping, Day 2
Factoring by Grouping
1) Group 2 terms together
and factor out GCF
2) Group remaining 2
terms and factor out GCF
3) Put the GCFs in a
binomial together
4) Put the common
binomial next to the GCF
binomial
Ex) 4qr + 8r + 3q + 6
Section 8-5: Grouping, Day 2
Factor the following by grouping
1) rn + 5n – r – 5
2) 3np + 15p – 4n – 20
Section 8-5: Grouping, Day 2
Factor by grouping with additive inverses.
1) 2mk – 12m + 42 – 7k
2)
c – 2cd + 8d – 4
Section 8-5: Zero Product Property,
Day 3
What is the point of factoring?
It is a method for solving nonlinear equations (quadratics,
cubics, quartics,…etc.)
Using ZPP:
1) Set equation equal to
__________.
2) Factor the non-zero side
Zero Product Property – If the
product of 2 factors is zero,
then at least one of the factors
MUST equal zero.
3) Set each __________
equal to ___________ and
solve for the variable
Section 8-5: Zero Product Property,
Day 3
Solve the equations using the ZPP
1) (x – 2)(x + 3) = 0
2) (2d + 6)(3d – 15) = 0
3)
c = 3c
2
4) 8b2 - 40b = 0
Section 8-6: Factoring Quadratics,
Day 1
Factoring quadratics in the form: ax 2 + bx + c = 0
Where a = 1, factors into 2 binomials:
(x + m)(x + n)
m + n = b the middle number in the trinomial
m x n = c the last number in the trinomial
Ex) x 2 + 7x +12  (x + 3)(x + 4)
Section 8-6: Factoring Quadratics,
Day 1
Factor the following trinomials
2
1) x + 6x +8
2) x 2 + 9x + 20
Section 8-6: Factoring Quadratics,
Day 1
Sign Rules:
ax + bx + c  (
+ )( + )
ax - bx + c  (
- )( - )
ax ± bx - c  (
+ )( - )
2
2
2
*If b is negative, the – goes with the bigger number
*If b is positive, the – goes with the smaller number
Section 8-6: Factoring Quadratics,
Day 1
Factor the following trinomials
1) x 2 - 8x +12
2) x 2 + 2x -15
3) x 2 - 7x -18
4) x 2 - 5x - 6
Section 8-6: Solving Quadratics by
Factoring, Day 2
Solve by factoring and using ZPP.
1) x 2 + 6x = 27
2) x 2 - 3x = 70
3) x 2 + 3x -18 = 0
4) x 2 + x = 20
Section 8-6: Solving Quadratics by
Factoring, Day 2
Word Problem: The width of a soccer field is 45
yards shorter than the length. The area is 9000
square yards. Find the actual length and width
of the field.
Section 8-7: The First/Last Method,
when a does not = 1, Day 1
First/Last Steps:
Ex)
1) Set up F, write factors of
the first number (a)
2) Set up L, write factors
of the last number (c)
3) Cross multiply. Can the
products add/sub to get
the middle number (b)? If
not, try new numbers for
F and L
2x + 5x + 3
2
Section 8-7: The First/Last Method,
when a does not = 1, Day 1
1) 7x 2 + 29x + 4
2) 3x 2 +15x +18
3) 4x 2 -13x +10
4) 2x 2 -17x + 30
Section 8-7: The First/Last Method,
when a does not = 1, Day 3
Factoring using First/Last when c is negative.
1) 3x 2 -11x - 20
2) 2x 2 - 3x - 9
Section 8-7: Factoring Completely,
Day 2
You must factor out a GCF FIRST! Then factor the
remaining trinomial into 2 binomials.
2
1) 10y - 35y + 30
2) 6x 2 + 22x - 8
Section 8-7: Solving by Factoring,
Day 2
Solve by factoring
1) 2x 2 + 9x + 9 = 0
2) -3x 2 + 26x =16
Section 8-7: Solving by Factoring,
Day 2
Lastly…Not all quadratics are factorable. These
are called PRIME. It does not mean they don’t
have a solution, it just means they cannot be
factored.
Ex) x 2 + 3x +11
Section 9-2: Solving Quadratics by
Graphing
Solutions of a Quadratic on a graph:
Section 9-2: Solving Quadratics by
Graphing
Solve the quadratics by graphing. Estimate the
solutions.
Ex) x 2 -10x +16 = 0
Section 9-2: Solving Quadratics by
Graphing
Solve the quadratics by graphing. Estimate the
solutions.
Ex) x 2 - 6x + 9 = 0
Section 9-2: Solving Quadratics by
Graphing
Solve the quadratics by graphing. Estimate the
solutions.
Ex) -2x 2 - 8x =13
Section 9-3: Transformations of
Quadratic Functions, Day 1
Transformation – Changes the position or size of
a figure on a coordinate plane
Translation – moves a figure up, down, left, or
right, when a constant k is added or subtracted
from the parent function
Section 9-3: Transformations of
Quadratic Functions, Day 1
Section 9-3: Transformations of
Quadratic Functions, Day 1
Describe how the graph of each function is
2
related to the graph of f (x) = x . First graph the
parent function, then graph the given function.
a) h(x) = x 2 + 3
b) g(x) = x 2 - 4
Section 9-3: Transformations of
Quadratic Functions, Day 1
Section 9-3: Transformations of
Quadratic Functions, Day 1
Describe how the graph of each function is
2
related to the graph of f (x) = x . First graph the
parent function, then graph the given function.
a) g(x) = (x - 2)2
b) h(x) = (x +1)2
Section 9-3: Transformations of
Quadratic Functions, Day 1
Describe how the graph of each function is
2
related to the graph of f (x) = x . First graph the
parent function, then graph the given function.
2
g(x)
=
(x
3)
+2
a)
b) g(x) = (x + 3)2 -1
Section 9-3: Transformations of
Quadratic Functions, Day 2
Section 9-3: Transformations of
Quadratic Functions, Day 2
Describe how the graph of each function is
2
related to the graph of f (x) = x . First graph the
parent function, then graph the given function.
1 2
2
h(x)
=
x
a)
b) g(x) = 3x + 2
2
Section 9-3: Transformations of
Quadratic Functions, Day 2
Section 9-3: Transformations of
Quadratic Functions, Day 2
Describe how the graph of each function is
2
related to the graph of f (x) = x . First graph the
parent function, then graph the given function.
a) g(x) = -2x 2 - 3
b) h(x) = -4(x + 2)2 +1
Section 9-3: Transformations of
Quadratic Functions, Day 2
Section 9-3: Transformations of
Quadratic Functions, Day 2
1
y = x2 - 4
3
1)
4) y = -3x 2 - 2
1
y = (x + 4)2 - 4
3
2)
5) y = -x 2 + 2
1 2
y= x +4
3
3)
6) y = (2x + 6)2 + 2
Section 9-3: Transformations of
Quadratic Functions, Day 2
f (x) = a(x - h)2 + k
Horizontal Translation (h) :
• If (x – h) move h spaces
to the right
Vertical Translation (k):
• If k is positive, move k
Spaces up
• If (x + h), move h
Spaces to the left
• If k is negative, move
k spaces down
Reflection (a)
• If a is positive, graph
Opens up
Dilation (a)
• If a is greater than 1,
There is a vertical stretch
(skinny)
• If a is negative, graph
Opens down
• If 0 < a < 1, there is a
Vertical compression
(fat)