# 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. 