Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663A/01) | Year 2014 | January | Q#4

Question

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

The curve crosses the y-axis at (0, 3) and has a minimum at P (4, 2).

On separate diagrams, sketch the curve with equation

a.   y = f(x + 4),

b.    y = 2f(x).

On each diagram show the coordinates of minimum point and any point of intersection with the y- axis.

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 4 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 ‘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 two-times of original given function.

It is shown in the figure below.