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 + 15

-4t + -4 = -4t – 4

- x2 – -9x = -x 2 + 9x

7n2 – 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 method2. 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

x 2

x

3

• Fill in the table with the product of each row and column value.

x2 2x

3x 6

• Write down the sum of all the products.

x2 + 2x + 3x + 6

• Simplify.x2 + 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

x

3

x2 -2x

3x -6

x2 + -2x + 3x + -6

x2 + 1x + -6

x2 + 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) =

FF

x 2 +

I

I

2x +

x2 + 5x + 6

• Pretty much the same as the table method…

3x +O

O

6L

L

some callit the claw

method

Fireworks – From Vertex to Standard Form

• FOIL Method (for multiplying binomials)

• How about one with subtraction…

(x + 4)·(x + –6) =

FF

x 2 +

x2 + –2x + –24

(x + 4)·(x – 6) = ?

x2 – 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 2 + 11x + 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: x2 – h2

• 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)2 + 7y = 2(x – 3)(x – 3) + 7

y = 2(x2 – 6x + 9) + 7y = (2x – 6)(x – 3) + 7

y = 2x2 – 12x + 18 + 7

y = 2x2 – 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)2 + 7

y = 2(x – 3)(x – 3) + 7

y = 2(x2 – 6x + 9) + 7

y = 2x2 – 12x + 18 + 7

y = 2x2 – 12x + 25

vertex form

standard form