A family of parabolas is defined by the quadratic function

MATHEMATICS · Multi-part · CBSE · Mathematics · Quadratic Equations · Quadratic Equations

Problem

y = 3(x - 2)^2 + p

, where

p

is a real constant. (i) Expand the expression to write the quadratic in the standard form

ax^2 + bx + c

and identify the values of

a

b

and

c

. (ii) For what range of values of

p

does the graph of

y = 3(x - 2)^2 + p

fail to intersect the x–axis? Show all steps of your reasoning. (iii) Sketch two separate graphs on the same set of axes: one for

p = 5

and the other for

p = -1

. Clearly indicate the vertex, axis of symmetry, and points of intersection with the x–axis (if any).