Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663/01)  Year 2012  June  Q#10
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Question
Figure 1 shows a sketch of the curve C with equation y = f(x) where
f (x) = x^{2}(9 –2x)
There is a minimum at the origin, a maximum at the point (3, 27) and C cuts the xaxis at the point A.
a. Write down the coordinates of the point A.
b. On separate diagrams sketch the curve with equation
i. y = f(x + 3)
ii. y = f(3x)
On each sketch you should indicate clearly the coordinates of the maximum point and any points where the curves cross or meet the coordinate axes.
The curve with equation y = f (x) + k, where k is a constant, has a maximum point at (3, 10).
c. Write down the value of k.
Solution
a.
We are given that the curve C cuts xaxis at point A, hence, A is xintercept of the given curve.
We are required to find coordinates of the point A.
The point at which curve (or line) intercepts xaxis, the value of . So we can find the value of coordinate by substituting in the equation of the curve (or line).
We are given that equation of the curve C is;
We substitute y=0 in this equation to find the coordinates of xintercept.
Now we have two options.







Hence, the curve C cuts xaxis at two points one where x=0 and other where . Since there is minimum point of the curve at origin, x=0, belongs to this point and belongs to point A.
Hence, coordinates of point A are .
b.
i.
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 3 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.
ii.
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 onethird of the original given function.
It is shown in the figure below.
c.
We are given that the curve with equation y = f (x) + k, where k is a constant, has a maximum point at (3, 10) and we are required to find the value of k.
We are given graph of y=f(x).
We are required to sketch y=f(x)+k.
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)+k is a case of translation by k 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)+k, we simply shift this y=f(x) graph k units along positive yaxis as. However, we are given that maximum point of the curve shifts from (3,27) to (3,10) which represents translation by 17 units along negative yaxis.
Therefore, k=17.
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