Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | 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 x-direction and  units in the  positive y-direction.

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 y-axis.

Original

Transformed

Translation

Vector

Movement

Function

 units in
positive y-direction

Coordinates

It is evident that y=f(x)+2 is a case of translation by 2 units along positive y-axis.

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.

To sketch y=f(x)+2, we simply shift this y=f(x) graph 2 units along positive y-axis 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
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 four-times 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
positive x-direction

 units in
positive y-direction

Coordinates

Function

 units in
negative x-direction

 units in
positive y-direction

Coordinates

Function

 units in
positive x-direction

 units in
negative y-direction

Coordinates

Function

 units in
negative x-direction

 units in
negative y-direction

Coordinates

Function

 units in
positive y-direction

Coordinates

Function

 units in
negative y-direction

Coordinates

Function

 units in
positive x-direction

Coordinates

Function

 units in
negative x-direction

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 x-direction.

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.

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

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 x-coordinates shifted 1 units in negative x-direction as  compared to that of original given function whereas all the y-coordinates 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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