Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2011 | June | Q#8

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Question

Figure 1 shows a sketch of the curve C with equation y = f (x).

The curve C passes through the origin and through (6, 0).

The curve C has a minimum at the point (3, –1).

On separate diagrams, sketch the curve with equation

a.   y=f(2x),

b.   y=-f(x)

c.   y=f(x+p), where p is a constant .

On each diagram show the coordinates of any points where the curve intersects the x-axis and of  any minimum or maximum points.

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 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 horizontal shrinking of the given  function.

Transformation of the function  into  results from shrinking of  in x- direction by a scale factor of .

Shrinking of the function  in x-direction by a scale factor of  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 the original given function  whereas all the x-coordinates are half of the original given function.

It is shown in the figure below. Orange graph is the original one whereas red represents the  transformed graph.

b.

We are given the sketch of the curve with equation;

We are required to sketch the curve of equation;

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

Reflection of the function  in x-axis transforms  into .

 Original Transformed Reflection in Function x-axis Coordinates

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

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

It is shown in the figure below. Orange graph is the original one whereas blue represents the  transformed graph.

c.

We are given the sketch of the curve with equation;

It is evident from the given diagram that minimum point of the graph is (3,-1).

We are required to sketch following function, where ;

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.

However, this graph may have any position shifted towards negative x-axis by p units.

The extremes are when p=1 (the green sketch) and when p=3 (the blue graph), in the figure below,  the actual graph for  will be somewhere between these extreme positions.