Lecture-17

Lecture-17 - June 24, 2010 Lecture 17: Quadratics The...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 perfect-square 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 u-v -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 u-v -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? 1 ...
View Full Document

This note was uploaded on 11/23/2011 for the course MATH 6302 taught by Professor Madden during the Summer '11 term at LSU.

Ask a homework question - tutors are online