Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663/01)  Year 2005  January  Q#6
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Question
Figure 1 shows a sketch of the curve with equation y = f(x). The curve crosses the xaxis at the points (2, 0) and (4, 0). The minimum point on the curve is P(3, –2).
In separate diagrams sketch the curve with equation
a. y=–f(x),
b. y=f(2x).
On each diagram, give the coordinates of the points at which the curve crosses the xaxis, and the coordinates of the image of P under the given transformation.
Solution
a.
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.
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 ‘stretch’ in transformation of given functions. Here .


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.
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