AS/A2 Pure 2 Pure 3

P1 Topic 2: Algebra
P1.1
Find the set of values of k for which x is real, given that y = x2+kx+(2k-3)=0 (a) For what values of k does the equation have: (i) equal roots (ii) roots of the same sign (iii) roots of opposite sign? (b) What is the significance of the common point on all the graphs? (c) What is the greatest minimum value that the expression y can have, and for what value of k does this occur? (d) Using your answers to (a) (i), can you find a curve which passes through the minimum point on each curve? (Hint: think about transformations of functions.) (e) Now prove all your answers on paper.

y = x2+kx+(2k-3)

y = x2+kx+(2k-3)    y = ax2+bx+c

Summary This problem is adapted from "Exploring Maths with Computers" by Aydin Önaç, The Chase, Malvern Based on free Java applets from JavaMath