Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663A/01)  Year 2014  January  Q#4
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Question
Figure 1 shows a sketch of a curve with equation y = f(x).
The curve crosses the yaxis at (0, 3) and has a minimum at P (4, 2).
On separate diagrams, sketch the curve with equation
a. y = f(x + 4),
b. y = 2f(x).
On each diagram show the coordinates of minimum point and any point of intersection with the y axis.
Solution
a.
We are given the sketch of the curve with equation;
We are required to sketch the curve of equation;
Translation through vector transforms the graph of into the graph of .
Transformation of the function into results from translation through vector .
Translation through vector represents the move, units in the positive xdirection and units in the positive ydirection.
Original 
Transformed 
Translation Vector 
Movement 

Function 



units in units in 
Coordinates 


However, for the given case we consider following.
Translation through vector represents the move, units in the negative xdirection and units in the ydirection.
Translation through vector transforms the function into .
Transformation of the function into results from translation through vector .
Translation through vector transforms the function into which means shift towards left along xaxis.
Original 
Transformed 
Translation Vector 
Movement 

Function 



units in 
Coordinates 


It is evident that we are required to transform the function into , therefore it is case of translation of along negative xaxis by 4 unit.
It is also evident from the above table that only xcoordinates of the graph change whereas y coordinates of the graph will remain unchanged.
Hence, the new function has all the ycoordinates same as that of original given function whereas all the xcoordinates are shifted towards negative xaxis of original given function.
It is shown in the figure below.
b.
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 twotimes of original given function.
It is shown in the figure below.
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