Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663/01)  Year 2008  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 (1, 0) and (4, 0). The maximum point on the curve is (2, 5).
In separate diagrams sketch the curves with the following equations.
On each diagram show clearly the coordinates of the maximum point and of each point at which the curve crosses the xaxis.
a. y=2f(x),
b. y=f(–x).
The maximum point on the curve with equation y=f(x + a) is on the yaxis.
c. Write down the value of the constant a.
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 
Value of 
Effect 

Function 


– 
‘Stretch’ 

Coordinates 



Function 



Expansion 

Coordinates 



Function 


Shrinking 

Coordinates 



Function 



Expansion 

Coordinates 



Function 


Shrinking 

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 .
Expansion of the function in ydirection by a scale factor of transforms into .
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 the original given function whereas all the ycoordinates will be double the 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;
Original 
Transformed 
Reflection 

Function 


xaxis 
Coordinates 



Function 


yaxis 
Coordinates 


It is evident that we are required to transform the function into , therefore it is case of reflection of in yaxis.
Transformation of the function into results from reflection of in yaxis.
Reflection of the function in yaxis 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 original given function whereas all the xcoordinates take opposite sign of original given function.
It is shown in the figure below.
c.
We are given the sketch of the curve with equation;
It is evident from the given diagram that maximum point of the graph is (2,5).
We are required to find the value of a in the following function equation when maximum point of the graph is on yaxis;
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.
It is evident that the maximum point of the graph will reach yaxis after it is translated 2 units towards negative xaxis of original given function.
Hence, .
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