Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663/01)  Year 2011  June  Q#8
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Question
Figure 1 shows a sketch of the curve C with equation y = f (x).
The curve C passes through the origin and through (6, 0).
The curve C has a minimum at the point (3, –1).
On separate diagrams, sketch the curve with equation
a. y=f(2x),
b. y=f(x)
c. y=f(x+p), where p is a constant .
On each diagram show the coordinates of any points where the curve intersects the xaxis and of any minimum or maximum points.
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 ‘stretch’ 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 horizontal shrinking of the given function.
Transformation of the function into results from shrinking of in x direction by a scale factor of .
Shrinking of the function in xdirection by a scale factor of transforms into .
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 the original given function whereas all the xcoordinates are half of the original given function.
It is shown in the figure below. Orange graph is the original one whereas red represents the transformed graph.
b.
We are given the sketch of the curve with equation;
We are required to sketch the curve of equation;
Transformation of the function into results from reflection of in xaxis.
Reflection of the function in xaxis transforms into .
Original 
Transformed 
Reflection 

Function 


xaxis 
Coordinates 


It is evident that we are required to transform the function into , therefore it is case of reflection of the given function in xaxis.
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 negative of original given function.
It is shown in the figure below. Orange graph is the original one whereas blue represents the transformed graph.
c.
We are given the sketch of the curve with equation;
It is evident from the given diagram that minimum point of the graph is (3,1).
We are required to sketch following function, where ;
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 a units.
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.
However, this graph may have any position shifted towards negative xaxis by p units.
The extremes are when p=1 (the green sketch) and when p=3 (the blue graph), in the figure below, the actual graph for will be somewhere between these extreme positions.
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