Complex numbers and polynomials - University College Cork

and Q has equationy = y 1 + y 2 − y 1x 2 − x 1(x − x 1 )= x 2 1 + x2 2 − x 2 1x 2 − x 1(x − x 1 )= x 2 1 + (x 2 + x 1 )(x − x 1 )= (x 2 + x 1 )x − x 1 x 2 .Thus, the line P Q lies above the arc ̂P Q sincefor all x between x 1 , x 2 , i.e.,x 2 − (x 2 + x 1 )x + x 1 x 2 = (x − x 1 )(x − x 2 ) ≤ 0,x 2 ≤ (x 2 + x 1 )x − x 1 x 2 ,for all x between x 1 , x 2 , as required.The square function separates the plane into two disjoint regions, viz., the sets ofpoints above and below its graph. The set above it is {(x, y) : y > x 2 }; the set belowis {(x, y) : y > x 2 }. The upper set is convex: the line segment joining any two of itspoints is wholly contained in it. The lower set is neither convex nor concave.The graph of any quadratic can be reduced to that of the square function or to itsreflection in the horizontal axis, by a process known as ‘completing the square’.To justify this, supposep(x) = ax 2 + bx + cis a quadratic polynomial, so that a ≠ 0. Then, if (x, y) is a point in its graph,y = ax 2 + bx + c= a(x 2 + b a x) + c= a(x 2 + b2a x + b24a ) + c − a b22 4a 2= a(x + b2a )2 +4ac − b2.4aThus4ac − b2y − = a(x + b4a2a )2 ,Or, changing the coordinate axes by translating the origin, we have Y = aX 2 , whereY = y −4ac − b2, X = x + b4a2a .Hence, the graph is convex, i.e., a smile if a > 0, and concave, i.e., a frown, if a < 0.8