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Unformatted text preview: June 24, 2010 Lecture 17: Quadratics
The standard form of a quadratic is
p(x) = a x2 + b x + c,
where x is a variable, and a, b, and c are constants, with a = 0. A quadratic polynomial in factored form is a(x − α)(x − β ), where a, α and β are constants.
This is transformed into standard form by using the distributive law repeatedly:
(x − α) (x − β ) = x2 − (α + β ) x + α β. An important instance of this is the perfectsquare identity:
x2 + 2 α x + α 2 = ( x + α ) 2 . (1) Completing the square
This refers to the following way of rewriting a quadratic, which is motivated by (1):
x2 + b x + c = x2 + 2 (b/2) x + (b/2)2 − (b/2)2 + c
2 = x + (b/2) − (b/2)2 − c . (2) We can apply (1) to the case of a quadratic with leading coeﬃcient a:
b
c
x+
a
a a x2 + b x + c = a x2 + 2 =a b
x+
2a
x+ b
2a 2 =a 2 − b
2a − c
a b2 − 4 a c
4a 2 Equation (3) shows that − (3) √ b2 − 4 a c
.
(4)
2a
We can also use equation (3) to select a uv coordinate system in which the equation
y = a x2 + b x + c has a simpler form. Let
b2 − 4 a c
b
, v=y+
.
(5)
u=x+
2a
4a
Then
y = a x2 + b x + c ⇔ v = a u2 .
2 ax + bx+c = 0 x= ⇔ −b ± The origin of the uv coordinate system is called the vertex of the graph of y = a x2 + b x + c.
It is at the point where:
−b2 + 4 a c
−b
, y=
.
(6)
x=
2a
4a
Problem. If we vary one of the coeﬃcients but hold the others ﬁxed, how does the graph
of y = a x2 + b x + c change? Does the shape of the graph change? How does the vertex
move? What happens to the y intercept?
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This note was uploaded on 11/23/2011 for the course MATH 6302 taught by Professor Madden during the Summer '11 term at LSU.
 Summer '11
 MADDEN
 Math

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