Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663/01)  Year 2016  June  Q#4
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Question
Figure shows a sketch of part of the curve with equation y = f(x). The curve has a maximum point A at (–2, 4) and a minimum point B at (3, –8) and passes through the origin O.
On separate diagrams, sketch the curve with equation
a. y = 3f(x),
b. y = f(x) – 4
On each diagram, show clearly the coordinates of the maximum and the minimum points and the coordinates of the point where the curve crosses the yaxis.
Solution
a.
We are given the sketch of the curve with equation;
We are required to sketch the curve of equation;
We know that and represent ‘stretched’ in transformation of given functions. Here , therefore;


Original 
Transformed 
Effect 

Function 


Expansion 
Coordinates 



Function 


Shrinking 
Coordinates 



Function 


Shrinking 
Coordinates 



Function 


Expansion 
Coordinates 


From the above table, as highlighted, it is evident that we are required to transform the function into , where , therefore it is case of vertical expansion of the given function.
Transformation of the function into results from expansion of in y direction by a scale factor of if .
Expansion of the function in ydirection by a scale factor of transforms into if .
It is also evident from the above table that only ycoordinates of the graph change whereas x coordinates of the graph will remain unchanged.
Hence, the new function has all the xcoordinates same as that of original given function whereas all the ycoordinates are threetimes of original given function.
It is shown in the figure below.
b.
We are given graph of y=f(x).
We are required to sketch y=f(x)4.
Translation through vector represents the move, units in the xdirection and units in the negative ydirection.
Translation through vector transforms the function into or .
Transformation of the function into or results from translation through vector .
Translation through vector transforms the function into or which means shift downwards along yaxis.
Original 
Transformed 
Translation Vector 
Movement 

Function 



units in 
Coordinates 


It is evident that y=f(x)4 is a case of translation by 4 units along negative yaxis.
It is also evident from the above table that only ycoordinates of the graph change whereas x coordinates of the graph will remain unchanged.
To sketch y=f(x) – 4, we simply shift this y=f(x) graph 4 units along negative yaxis as shown below.
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