Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2008 | January | Q#6

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Question

Figure 1 shows a sketch of the curve with equation y= f(x). The curve crosses the x-axis at the  points (1, 0) and (4, 0). The maximum point on the curve is (2, 5).

In separate diagrams sketch the curves with the following equations.

On each diagram show clearly the coordinates of the maximum point and of each point at which the  curve crosses the x-axis.

a.   y=2f(x),

b.   y=f(–x).

The maximum point on the curve with equation y=f(x + a) is on the y-axis.

c.   Write down the value of the constant a.

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 ‘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.

b.

We are given the sketch of the curve with equation;

We are required to sketch the curve of equation;

 Original Transformed Reflection in Function x-axis Coordinates of Points Function y-axis Coordinates of Points

It is evident that we are required to transform the function  into , therefore it is case of reflection of  in y-axis.

Transformation of the function  into  results from reflection of  in y-axis.

Reflection of the function  in y-axis transforms  into

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 take opposite sign of original given function.

It is shown in the figure below.

c.

We are given the sketch of the curve with equation;

It is evident from the given diagram that maximum point of the graph is (2,5).

We are required to find the value of a in the following function equation when maximum point of the  graph is on y-axis;

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 a units.

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 evident that the maximum point of the graph will reach y-axis after it is translated 2 units  towards negative x-axis of original given function.

Hence, .