Fireworks – From Vertex to Standard Form The distributive property of multiplication. a·(b + c)a·b+a·c= Examples 3·(x + 5) - 4·(t + 1) - x·(x – 9) 7n·(n – 2) = = = = 3x t + - 4= - 4t – 4 - x 2 – - 9x= - x 2 + 9x 7n 2 – 14n

Fireworks – From Vertex to Standard Form The distributive property of multiplication (cont'd) Things get a little more complicated in situations like… (a + b)·(c + d) Basically, each of the two terms in the first binomial have to be multiplied by each of the two terms in the second binomial. We will be looking at two methods for doing this kind of multiplication. 1.Table method 2.FOIL method Both are methods of DOUBLE DISTRIBUTION

Fireworks – From Vertex to Standard Form Table Method (for multiplying binomials) (x + 2)·(x + 3) = ? Create a 2-by-2 table Put a "+" at the top divider and another at the side divider + + Place two terms of first binomial at top and the second's on the side x2 x 3 Fill in the table with the product of each row and column value. x2x2 2x2x 3x3x6 Write down the sum of all the products. x 2 + 2x + 3x + 6 Simplify. x 2 + 5x + 6

Fireworks – From Vertex to Standard Form Table Method (for multiplying binomials) In cases where you have a subtraction, change it to "adding a negative" and follow same steps. (x – 2)·(x + 3) (x + - 2)·(x + 3) + + x -2-2 x 3 x2x2 -2x-2x 3x3x -6-6 x x + 3x x 2 + 1x x 2 + x – 6

Fireworks – From Vertex to Standard Form FOIL Method (for multiplying binomials) FOIL stands for First Outer Inner Last Here's how it works… (x + 2)·(x + 3) = F F x 2 + I I 2x + x 2 + 5x + 6 Pretty much the same as the table method… 3x + O O 6 L L some call it the claw method

Fireworks – From Vertex to Standard Form FOIL Method (for multiplying binomials) How about one with subtraction… (x + 4)·(x + – 6) = F F x 2 + x 2 + – 2x + – 24 (x + 4)·(x – 6) = ? x 2 – 2x – 24 4x + I I – 6x + O O – 24 L L

Fireworks – From Vertex to Standard Form Practice finding products using a method of your choice. 1.)(x + 9)(x + 2) = 2.)(x – 3)(x + 10) = 3.)(x – 8)(x – 6) = x x + 18 x 2 + 7x – 30 x 2 – 14x + 48

Fireworks – From Vertex to Standard Form As we know, the vertex form of a quadratic equation is y = a·(x – h) 2 + k Using the table or FOIL method, we can deal with the (x – h) 2 part of the equation. IMPORTANT:x 2 – h 2 When raising any amount to the second power, we multiple the amount by itself. (x – h) 2 (x – h)(x – h) (x – h) 2 … use the table or FOIL method to finish it off.

Fireworks – From Vertex to Standard Form Putting it all together… y = 2(x – 3) y = 2(x – 3)(x – 3) + 7 y = 2(x 2 – 6x + 9) + 7 y = (2x – 6)(x – 3) + 7 y = 2x 2 – 12x y = 2x 2 – 12x + 25 vertex form standard form (x – 3)(x – 3)2(x – 3)

Fireworks – From Vertex to Standard Form Putting it all together… y = 2(x – 3) y = 2(x – 3)(x – 3) + 7 y = 2(x 2 – 6x + 9) + 7 y = 2x 2 – 12x y = 2x 2 – 12x + 25 vertex form standard form