AS/A2 Pure 1 Pure 3

P2 Topic 2: Functions
P2.2
(a) Compare the graphs of y = ax2+bx+1, y = |ax2+bx+1| and y = a|x2|+b|x|+1, for various values of a and b. How are the curves related? Now prove your answers on paper. (b) Compare the graphs of y = a|x2|+b|x|+1 and y = a(x2)2+bx2+1, for various values of a and b. How are they related? Prove your answer. (c) Investigate the graphs of y = f(|x|) and y =f(x2) for various functions f(x). Use the constants a and b in your functions. To use the modulus function |x|, you must enter abs(x).

y = ax2+bx+1, y = |ax2+bx+1|, y = a|x2|+b|x|+1

y = a|x2|+b|x|+1, y = a(x2)2+bx2+1

y = a|x|+b, y = ax²+b

Summary This problem is adapted from "A Level Investigations with Omnigraph" Based on free Java applets from JavaMath