1. 1) Find f(-3),f(-2),f(-1), f(0), f(1), f(2), f(3) 2) Sketch the graph
Algebra ,10 May 2011
Warm up: Evaluate f(x) = x2 + 2x - 3
1) Find f(-3),f(-2),f(-1), f(0), f(1), f(2), f(3)
2) Sketch the graph
3) Identify the roots and the vertex
4) for what x values is f(x) = 0
5) graph on the calculator
PROJECT DUE May 11

2. objectives
Students will multiply binonials and factor trinomials. Students will take notes, participate in class discussion and work with their group to solve problems.

3. quadratic equations
STANDARD FORM: y = f(x) = ax2 + bx + c VERTEX FORM: y = f(x) = a(x – h)2 + k Factored form: y = f(x) = (x – n)(x – m)

4. from factored form to standard form
y = (x – n)(x – m)
 y = ax2 + bx + c
HOW? Multiply the binomials
and combine like terms!!
like terms- same variable(s) to the same power(s)

5. LIKE TERMS have the same variables raised to the same power Find f(2) and f(5) for each function 1) f(x)= 2x + 1 2) f(x) = 3x DOES 2x + 1 = 3x? NO! 3) f(x) = 2x2+ x 4) f(x) = 3x2 DOES 2x2+ x = 3x2? NO!! We can only combine (add or subtract) LIKE TERMS!!

6. use an area model Rewrite y = (x + 2)(x + 3) in standard form x + 2
x(x) = x2 2x + 3x
3(2) = 6 3
y =(x + 2)(x + 3) = x2 + 2x + 3x + 6
y = x2 + 5x + 6

7. Or use FOIL or “double rainbow” then combine like terms
(X + 2)(X + 3) = X2 + 3X + 2X + 6 = X2 + 5X + 6 I L
Multiply in this order:
F- First terms
O- Outer terms
I- Inner terms
L- Last terms

8. Multiplying Binomials (FOIL)
Multiply. (x+3)(x+2)
Distribute.
x • x + x • • x + 3 • 2 F O I L
= x2+ 2x + 3x + 6
= x2+ 5x + 6

9. Multiplying Binomials (Tiles)
Multiply. (x+3)(x+2)
Using Algebra Tiles, we have: x x2 x x x x + 2
= x2 + 5x + 6 x 1 1 1 x 1 1 1

10. Finish FOIL/ Area Model Handout
Be ready to discuss in 10 minutes

11. Factoring Trinomials (Tiles)
How can we factor trinomials such as x2 + 7x + 12?
How can we change the quadratic from standard form to factored form?
One method is to again use algebra tiles– we need to make a rectangle: x2 x x x x x
1) Start with x2. x
2) Add seven “x” tiles (vertical or horizontal, at least one of each) and twelve “1” tiles. 1 1 1 1 1 x 1 1 1 1 1 1 1

12. Factoring Trinomials (Tiles)
How can we factor trinomials such as
x2 + 7x + 12 back into binomials?
One method is to again use algebra tiles:
1) Start with x2. x2 x x x x x
2) Add seven “x” tiles (vertical or horizontal, at least one of each) and twelve “1” tiles. x 1 1 1 1 1 x 1 1 1 1 1 1 1
3) Rearrange the tiles until they form a rectangle!
We need to change the “x” tiles so the “1” tiles will fill in a rectangle.

13. Factoring Trinomials (Tiles)
How can we factor trinomials such as
x2 + 7x + 12 back into binomials?
One method is to again use algebra tiles:
1) Start with x2. x2 x x x x x x
2) Add seven “x” tiles (vertical or horizontal, at least one of each) and twelve “1” tiles. x 1 1 1 1 1 1 1 1 1 1 1 1
3) Rearrange the tiles until they form a rectangle!
Still not a rectangle.

14. Factoring Trinomials (Tiles)
How can we factor trinomials such as
x2 + 7x + 12 back into binomials?
One method is to again use algebra tiles:
1) Start with x2. x2 x x x x
2) Add seven “x” tiles (vertical or horizontal, at least one of each) and twelve “1” tiles. x 1 1 1 1 x 1 1 1 1 x 1 1 1 1
3) Rearrange the tiles until they form a rectangle!
A rectangle!!!

15. Factoring Trinomials (Tiles)
How can we factor trinomials such as
x2 + 7x + 12 back into binomials?
One method is to again use algebra tiles: x
+ 4
4) Top factor: The # of x2 tiles = x’s The # of “x” and “1” columns = constant. x2 x x x x x + 3 x 1 1 1 1 x 1 1 1 1
5) Side factor: The # of x2 tiles = x’s The # of “x” and “1” rows = constant. x 1 1 1 1
x2 + 7x + 12 = ( x + 4)( x + 3)

16. X- Box
Product 3 -9
Sum

17. First and Last Coefficients
Factor the x-box way
y = ax2 + bx + c
Base 1
Base 2
Product
ac=mn
First and Last Coefficients
1st Term
Factor n
GCF n m
Middle
Last term
Factor m
b=m+n
Sum
Height

18. Examples Factor using the x-box method.
1. x2 + 4x – what two numbers multiply
to – 12 AND add to + 4??
a) b) x +6
-12 4
x x
-2x x 6 -2 -2
Solution: x2 + 4x – 12 = (x + 6)(x - 2)
Standard form Factored form
a = 1 b = 4 c = -12

19. Examples continued
2. x2 - 9x what two numbers multiply to 20 and also add to – 9?
a) b) x -4 20 -9 x
x2 -4x
-5x 20 -5
Solution: x2 - 9x + 20 = (x - 4)(x - 5)
standard form factored form
a = ? b = ? c = ?

20. practice FOIL/ area model
Do problems on handout
FACTORING– THREE METHODS
Area Model
x-method
Find two factors that multiply
to get the last number (when a = 1)
and
also add to get the middle number

21. exit quiz
Multiply using FOIL and an area model A B (x + 2)(x + 6) (x + 3)(x + 5) C D (x + 4)(x + 3) (x + 2)(x + 4) FACTOR using your favorite method: x2 + 10x + 9 x2 + 6x + 8 A B C D