Let be a polynomial function where and .

MATHEMATICS · Multi-part · GCSE · Mathematics · Graphs · Graphs

Problem

Let

p(x)=a(x-1)^{4}+b

be a polynomial function where

a\neq0

and

a,\,b\in\mathbb{R}

. (i) For what sign of

a

does the function attain a greatest value rather than a least value? Give a reason. (ii) Assuming

a<0

, write the domain and the range of

p

in terms of

a

and

b

. (iii) If

a=-2

and the greatest value of

p

5

, find the value of

b

and state the complete range of the function in interval notation. (iv) Sketch the graph of

y=p(x)

for the values of

a

and

b

obtained in part (iii), indicating the vertex and the intercepts with the axes.