1) Find f(-3),f(-2),f(-1), f(0), f(1), f(2), f(3) 2) Sketch the graph Algebra 1 9,10 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

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

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)

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)

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!!

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

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

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

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

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

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

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.

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.

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!!!

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)

X- Box Product 3 -9 Sum

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

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

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

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

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