Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2006 | January | Q#6

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Question

Figure 1 shows a sketch of the curve with equation y=f(x). The curve passes through the points  (0,3) and (4,0) and touches the x-axis at point (1,0).

On separate diagrams sketch the curve with equation

a.   y=f(x+1),

b.   y=2f(x),

c.  

On each diagram show clearly the coordinates of all the points where the curve meets the axes.

Solution

a.
 

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 x-direction and  units in
the positive y-direction.

Original

Transformed

Translation

Vector

Movement

Function

 units in
positive x-direction

 units in
positive y-direction

Coordinates

However, for the given case we consider following.

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

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 x-axis.

Original

Transformed

Translation

Vector

Movement

Function

 units in
negative x-direction

Coordinates

It is evident that we are required to transform the function  into , therefore it is  case of translation of  along negative x-axis by 1 unit.

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

Hence, the new function has all the y-coordinates same as that of original given function whereas  all the x-coordinates are shifted towards negative x-axis of 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;

We know that  and  represent ‘stretch’ in transformation of given functions. Here  , therefore;

Original

Transformed

Value of

Effect

Function

‘Stretch’
Vertically by

Coordinates

Function

Expansion
Vertically by

Coordinates

Function

Shrinking
Vertically by

Coordinates

Function

Expansion
Vertically by

Coordinates

Function

Shrinking
Vertically 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 .

Expansion of the function  in y-direction by a scale factor of  transforms  into  .

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 the original given function  whereas all the y-coordinates will be double the original given function.

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  and  represent ‘stretch’ in transformation of given functions. Here  , therefore;

Original

Transformed

Value of

Effect

Function

‘Stretch’
Horizontally by

Coordinates

Function

Shrinking
Horizontally by

Coordinates

Function

Expansion
Horizontally by

Coordinates

Function

Shrinking
Horizontally 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 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 x-direction by a scale factor of  transforms  into   if .

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

Hence, the new function has all the y-coordinates same as that of the original given function  whereas all the x-coordinates will be ‘c’ times the original given function.

It is shown in the figure below.

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