Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663/01R)  Year 2013  June  Q#9
Question
Figure 1 shows a sketch of the curve C with equation y = f(x).
The curve C passes through the point (1,0) and touches the xaxis at the point (2,0).
The curve C has a maximum at the point (0,4).
a. The equation of the curve C can be written in the form
Where a, b and c are integers.
Calculate the values of a, b and c.
b. Sketch the curve with equation in the space provided on page 24.
Show clearly the coordinates of all the points where the curve crosses or meets the coordinate axes.
Solution
a.
We are given the sketch of the curve C with equation y = f(x). It is evident that it is a cubic graph and it is evident from the given general equation of the curve, as well.
If is a polynomial of degree then will have exactly factors, some of which may repeat.
A cubic equation can be written as a product of its all three factors as follows;
This is case where all three roots of the cubic polynomial are distinct. This means the cubic graph crosses the xaxis at three distinct points with coordinates and .
However, if cubic graph has a repeated roots and one distinct roots then;
In this case, one factor is repeated and other is distinct. This means the cubic graph crosses the xaxis at two distinct points with coordinates and . The repeated root shows that the cubic graph only touches the xaxis at this point .
From the given sketch of the curve C we can see that graph crosses xaxis at point (1,0) and touches it at point (2,0).
Hence, factors of the cubic graph are and repeated .
Therefore;
We can expand the right hand side, to equate with the left hand side.
Comparing both sides yields that;
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 , therefore;
Original 
Transformed 
Value of 
Effect 

Function 


– 
‘Stretch’ 
Coordinates 



Function 



Shrinking 
Coordinates 



Function 


Expansion 

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 expansion of the given function.
Transformation of the function into results from expansion of in x direction by a scale factor of if .
Expansion of the function in xdirection by a scale factor of transforms into if .
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 will be ‘c’ times the original given function.
It is shown in the figure below.
The red graph is the original one whereas orange graph represents the transformed one.
It is evident that only xcoordinates of the graph are times the original ones whereas ycoordinates remain unchanged.
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