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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