Consider the function 

1. State, in terms of where required, the values of for which .  (1 mark
   

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2. Find the values of for which the graph of has
     
    i. exactly three -intercepts.   (2 marks)
   

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   ii. exactly four -intercepts.   (1 mark)
   

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3. Let be the function , which is the function where .
     
     i. Find    (1 mark)
   

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    ii. Find the coordinates of the local maximum of .   (1 mark)
   

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   iii. Find the values of for which .   (1 mark)
   

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    iv. Consider the two tangent lines to the graph of at the points where
   and . Determine the coordinates of the point of intersection of these two tangent lines.   (2 marks)

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4. Let remain as the function , which is the function where .

   Let be the function , which is the function where .
     
    i. Using translations only, describe a sequence of transformations of , for which its image would have a local maximum at the same coordinates as that of .   (1 mark)
   

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   ii. Using a dilation and translations, describe a different sequence of transformations of , for which its image would have both local minimums at the same coordinates as that of .   (2 marks)
   

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