Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663/01)  Year 2010  January  Q#8
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Question
Figure 1 shows a sketch of part of the curve with equation y = f(x). The curve has a maximum point (−2, 5) and an asymptote y = 1, as shown in Figure 1.
On separate diagrams, sketch the curve with equation
a. y = f(x) + 2
b. y = 4f(x)
c. y = f(x + 1)
On each diagram, show clearly the coordinates of the maximum point and the equation of the asymptote.
Solution
a.
We are given graph of y=f(x).
We are required to sketch y=f(x)+2.
Translation through vector represents the move, units in the xdirection and units in the positive ydirection.
Translation through vector transforms the function into or .
Transformation of the function into or results from translation through vector .
Translation vector transforms the function into or which means shift upwards along yaxis.
Original 
Transformed 
Translation Vector 
Movement 

Function 



units in 
Coordinates 


It is evident that y=f(x)+2 is a case of translation by 2 units along positive yaxis.
It is also evident from the above table that only xcoordinates of the graph change whereas y coordinates of the graph will remain unchanged.
To sketch y=f(x)+2, we simply shift this y=f(x) graph 2 units along positive yaxis as and same applies to asymptote as well, as shown 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 fourtimes of original given function. The same applies to asymptote as well.
It is shown in the figure below.
c.
We are given the sketch of the curve with equation;
We are required to sketch the curve of equation;
We know that represent ‘translation’ in transformation of given functions. Here and , therefore;
Original 
Transformed 
Translation Vector 
Movement 

Function 



units in units in 
Coordinates 



Function 



units in units in 
Coordinates 



Function 



units in units in 
Coordinates 



Function 



units in units in 
Coordinates 



Function 



units in 
Coordinates 



Function 



units in 
Coordinates 



Function 



units in 
Coordinates 



Function 



units in 
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 translation of the given function in negative xdirection.
Transformation of the function into results from translation through vector .
Translation through vector transforms the function into which means shift towards left along xaxis.
Translation through vector represents the move, units in the negative xdirection and units in the ydirection.
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 xcoordinates shifted 1 units in negative xdirection as compared to that of original given function whereas all the ycoordinates are remain same as that of original given function. The same applies to asymptote as well.
It is shown in the figure below.
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