Mathematics

Problem

The graphs of three quadratic polynomials

y=p_1(x)

y=p_2(x)

and

y=p_3(x)

exhibit the following features in the

xy

–plane: (A) The curve cuts the

x

–axis at two distinct points. (B) The curve touches the

x

–axis at exactly one point and turns back. (C) The curve remains entirely above the

x

–axis and never meets it. a. (i) For each of the cases (A), (B) and (C), state the exact number of distinct real zeros of the corresponding polynomial. b. (ii) Denote the discriminants of the three polynomials by

D_A, D_B

and

D_C

respectively. Arrange

D_A, D_B

and

D_C

in increasing order, giving reasons for your answer. c. (iii) Explain, with justification, how the position of the vertex of a parabola with respect to the

x

–axis helps in deciding the number of its real zeros.

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