Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2016 | June | Q#4

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Question

Figure  shows a sketch of part of the curve with equation y = f(x). The curve has a maximum point A  at (–2, 4) and a minimum point B at (3, –8) and passes through the origin O. 

On separate diagrams, sketch the curve with equation

a.   y = 3f(x),

b.   y = f(x) – 4

On each diagram, show clearly the coordinates of the maximum and the minimum points and the  coordinates of the point where the curve crosses the y-axis.

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 ‘stretched’ in transformation of given functions.  Here , therefore;

Original

Transformed

Effect

Function

Expansion
Vertically by

Coordinates

Function

Shrinking
Horizontally by

Coordinates

Function

Shrinking
Vertically by

Coordinates

Function

Expansion
Horizontally by

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 y-direction by a scale factor of  transforms  into   if .

It is also evident from the above table that only y-coordinates of the graph change whereas x- coordinates of the graph will remain unchanged. 

Hence, the new function has all the x-coordinates same as that of original given function whereas  all the y-coordinates are three-times of original given function. 

It is shown in the figure below.

b.
 

We are given graph of y=f(x).

We are required to sketch y=f(x)-4.

Translation through vector  represents the move,  units in the x-direction and   units in the negative y-direction.

Translation through vector  transforms the function  into  or

Transformation of the function  into  or  results from translation  through vector .

Translation through vector  transforms the function  into  or  which means shift downwards along y-axis.

Original

Transformed

Translation

Vector

Movement

Function

 units in
negative y-direction

Coordinates

 

It is evident that y=f(x)-4 is a case of translation by 4 units along negative y-axis. 

It is also evident from the above table that only y-coordinates of the graph change whereas x- coordinates of the graph will remain unchanged.

To sketch y=f(x) – 4, we simply shift this y=f(x) graph 4 units along negative y-axis as shown below. 

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